# Monte Carlo convergence rates for $k$th moments in Banach spaces

Kristin Kirchnera,∗ and Christoph Schwabb $\,{}^{{a}}$Delft Institute of
Applied Mathematics, Delft University of Technology
P.O. Box 5031, 2600 GA Delft, The Netherlands
$\,{}^{*}$Corresponding author; e-mail<EMAIL_ADDRESS>$\,{}^{{b}}$Seminar for Applied Mathematics, ETH Zürich
Rämistrasse 101, 8092 Zürich, Switzerland

###### Abstract.

We formulate standard and multilevel Monte Carlo methods for the $k$th moment
$\mathbb{M}^{k}_{\varepsilon}[\xi]$ of a Banach space valued random variable
$\xi\colon\Omega\to E$, interpreted as an element of the $k$-fold injective
tensor product space $\otimes^{k}_{\varepsilon}E$. For the standard Monte
Carlo estimator of $\mathbb{M}^{k}_{\varepsilon}[\xi]$, we prove the
$k$-independent convergence rate $1-\tfrac{1}{p}$ in the
$L_{q}(\Omega;\otimes^{k}_{\varepsilon}E)$-norm, provided that (i) $\xi\in
L_{kq}(\Omega;E)$ and (ii) $q\in[p,\infty)$, where $p\in[1,2]$ is the
Rademacher type of $E$. By using the fact that Rademacher averages are
dominated by Gaussian sums combined with a version of Slepian’s inequality for
Gaussian processes due to Fernique, we moreover derive corresponding results
for multilevel Monte Carlo methods, including a rigorous error estimate in the
$L_{q}(\Omega;\otimes^{k}_{\varepsilon}E)$-norm and the optimization of the
computational cost for a given accuracy. Whenever the type of the Banach space
$E$ is $p=2$, our findings coincide with known results for Hilbert space
valued random variables.

We illustrate the abstract results by three model problems: second-order
elliptic PDEs with random forcing or random coefficient, and stochastic
evolution equations. In these cases, the solution processes naturally take
values in non-Hilbertian Banach spaces. Further applications, where physical
modeling constraints impose a setting in Banach spaces of type $p<2$, are
indicated.

###### Key words and phrases:

Banach space valued random variable, injective tensor product, Monte Carlo
estimation, multilevel methods, Rademacher averages, type of Banach space

###### 2010 Mathematics Subject Classification:

Primary 65C05; Secondary 46A32, 60B11, 60H35.

##  1\. Introduction

###  1.1. Background and motivation

Many applications in uncertainty quantification require the estimation of
statistical moments. The first statistical moment, that is to say the mean, is
often itself the quantity of interest, whereas higher-order moments are needed
to infer certain characteristics about the probability distribution of the
underlying real- or vector-valued random variable. In the case that this
distribution is Gaussian, it is fully determined by the first two statistical
moments. Third-order and fourth-order moments, which define the skewness and
kurtosis of the probability distribution, play for instance an important role
for tests if the distribution is Gaussian, see e.g. [38, 49].

In order to estimate statistical moments one resorts to sampling strategies,
i.e., Monte Carlo methods. It is well-known that for estimating the mean the
convergence rate $\nicefrac{{1}}{{2}}$ in the number of samples is achieved as
long as the random variable $\xi$ is square-integrable in Bochner sense with
values in a Hilbert space $H$, i.e., $\xi\in L_{2}(\Omega;H)$. Moreover, this
result extends to statistical moments of an arbitrary order $k\in\mathbb{N}$
when interpreted as elements of the Hilbert tensor product space $H^{(k)}$,
provided that the random variable exhibits sufficient integrability in
$L_{2k}(\Omega;H)$.

Vector-valued random variables occur, for instance, in the context of
differential equations involving randomness. Here, numerical methods for
generating samples of approximate solutions often allow for a hierarchical
multilevel structure corresponding to different degrees of refinement of the
discretization parameters. The idea of multilevel Monte Carlo (MLMC) methods
is to reduce the computational cost for achieving a given target accuracy by
optimizing the number of samples used on each level to compute the MLMC
estimator. To the best of our knowledge this approach was first formulated by
Giles [22] for stochastic ordinary differential equations (SDEs) after having
previously been introduced by Heinrich [32] in the context of numerical
integration. Since then MLMC methods have been used to approximate means of
Hilbert space valued random variables for a variety of problems in uncertainty
quantification, including but not limited to SDEs [13, 21, 23, 25, 58],
partial differential equations (PDEs) with random coefficients [3, 10, 12, 13,
26, 27, 31, 56], stochastic PDEs [2, 24], and hyperbolic PDEs with random
fluxes or uncertainties in the initial data [50, 51, 52].

With regard to higher-order moments, MLMC strategies have been applied to
estimate _diagonals of central statistical moments_ in [5, 6], and combined
with sparse tensor techniques to approximate (_full_) moments in [3, 51, 52];
the latter approach has been refined and generalized by the multi-index Monte
Carlo method in [30].

All of the previously mentioned references have in common that the error
analysis of the (multilevel) Monte Carlo estimators proceeds in Hilbert
spaces. For the first statistical moment, it is known that estimating means of
random variables taking values in a Banach space $E$ via the standard Monte
Carlo method _does in general not converge at the rate $\nicefrac{{1}}{{2}}$_,
even in the presence of high Bochner integrability. More specifically, the
rate of convergence depends on geometric properties of the Banach space $E$:
If $E$ has _Rademacher type_ $p\in[1,2]$ and the random variable is an element
of the Bochner space $L_{q}(\Omega;E)$, $q\geq 1$, the standard Monte Carlo
method will, in general, converge only at the rate
$1-\tfrac{1}{\min\\{p,q\\}}$, see [47, Proposition 9.11]. This behavior
necessitates tailoring of MLMC methods not only to the discretization of a
particular problem, but also to the type of the Banach space. The only two
references known to the authors addressing this issue are [42], where a
scalar, degenerate conservation law with random data is discretized by a MLMC
finite difference method, and [14], where the authors perform a MLMC analysis
in Hölder spaces for solutions to stochastic evolution equations.

Besides the aforementioned issue of type-dependent convergence rates, another
difficulty occurs when considering higher-order moments of Banach space valued
random variables: As opposed to the Hilbert space case, there is no canonical
choice for the norm on the tensor product space $E\otimes E$. Two options
which are widely used in the literature are the _projective_ and _injective
tensor product norms_ , mostly caused by the fact that any reasonable cross
norm is bounded from above, respectively from below, by these norms, see [54,
Proposition 6.1(a)]. Janson and Kaijser [37] defined and analyzed statistical
moments of order $k\in\mathbb{N}$ (in Bochner, Dunford or Pettis sense) as
elements of projective and injective tensor product spaces. One of the
findings [37, Theorem 3.8] shows that both the _projective_ and _injective
$k$th moment_ of $\xi\colon\Omega\to E$ exist in Bochner sense (and coincide)
whenever $\xi\in L_{k}(\Omega;E)$.

Clearly, the choice of the tensor product norm will play a crucial role in the
error analysis of (multilevel) Monte Carlo estimation of higher-order
statistical moments. A simple argumentation (see Example 3.21) shows that, no
matter how “good” the (Rademacher) type of the Banach space $E$ is, Monte
Carlo methods for the second moment will in general not converge in the
projective tensor product of $E$.

While this work is devoted to the numerical analysis of Monte Carlo (sampling)
methods to approximate higher-order moments of vector-valued random variables,
we remark that certain _linear_ (or _linearized_) stochastic equations allow
for alternative approaches to _deterministically compute_ approximations to
$k$-point correlations of random solutions. In this context, we mention [11,
40, 41, 43, 46] and the references therein. This methodology does not raise
the mathematical issue of the type of a Banach space and its impact on the
convergence of numerical approximations. In the present paper, we shall not
pursue this direction further.

###  1.2. Contributions

We consider the injective $k$th moment $\mathbb{M}^{k}_{\varepsilon}[\xi]$ of
a Banach space valued random variable $\xi\in L_{k}(\Omega;E)$ and, for the
first time, formulate standard and multilevel Monte Carlo methods for this
higher-order moment in the Banach space setting. We prove that the standard
Monte Carlo estimator for $\mathbb{M}^{k}_{\varepsilon}[\xi]$ converges in the
$L_{q}(\Omega;\otimes^{k}_{\varepsilon}E)$-norm at the rate $1-\tfrac{1}{p}$
provided that (i) $\xi\in L_{kq}(\Omega;E)$ and (ii) $q\in[p,\infty)$, where
$p\in[1,2]$ is the Rademacher type of $E$, see Theorem 3.16. Here,
$\otimes^{k}_{\varepsilon}E$ denotes the $k$-fold injective tensor product of
$E$. Note, in particular, thatthis convergence rate is independent of the
order $k$ of the statistical moment $\mathbb{M}^{k}_{\varepsilon}[\xi]$. This
result readily implies error estimates and convergence rates for abstract
_single-level Monte Carlo_ methods for $\mathbb{M}^{k}_{\varepsilon}[\xi]$,
see Corollary 3.20.

By means of replacing Rademacher sums by Gaussian averages and exploiting a
version of Slepian’s inequality for Gaussian processes due to Fernique [18],
we are furthermore able to formulate corresponding abstract results for
_multilevel Monte Carlo_ methods. This includes a rigorous error estimate in
the $L_{q}(\Omega;\otimes^{k}_{\varepsilon}E)$-norm, see Theorem 3.24, and the
optimization of the computational cost for a given accuracy (in the MLMC
context also known as “$\alpha\beta\gamma$ theorem”), see Theorem 3.25.

We apply these abstract results to several classes of problems, where the
stochastic solution processes naturally take values in (non-Hilbertian) Banach
spaces: second-order elliptic PDEs  (a) with random forcing taking values in
$L_{p}$ for some general $p\in(1,\infty)$, or  (b) with log-Gaussian diffusion
coefficient and right-hand side in $L_{p}$; and  (c) stochastic evolution
equations, where we extend the MLMC analysis in Hölder norms, performed in
[14, Section 5] for mean values of solution processes, to their $k$th moments,
being $k$th order (spatio-)temporal correlation functions.

###  1.3. Layout

In Section 2 we introduce the necessary notation, see Subsection 2.1, as well
as the analytical preliminaries on tensor products of a Banach space $E$, with
particular emphasis on the full and symmetric $k$-fold injective tensor
product of $E$, see Subsection 2.2. We close this section with the definition
of the injective $k$th moment of a Banach space valued random variable $\xi\in
L_{k}(\Omega;E)$ in Subsection 2.3. Section 3 is dedicated to the analysis of
Monte Carlo methods for the injective $k$th moment. For this purpose, we first
need to formulate several auxiliary results for Rademacher and Gaussian
averages in Subsection 3.1. We then perform the error analysis for the
standard (and single-level) Monte Carlo method in Subsection 3.2 and for the
multilevel Monte Carlo method in Subsection 3.3. In Section 4 we discuss
several applications of our convergence results. Section 5 gives an outlook on
extensions and further applications, where due to essential restrictions in
the physical modeling a (non-Hilbertian) Banach space setting cannot be
avoided.

##  2\. Preliminaries

###  2.1. General notation and setting

Given parameter sets $\mathscr{P},\,\mathscr{Q}$, and mappings
$\mathscr{F},\mathscr{G}\colon\mathscr{P}\times\mathscr{Q}\to\mathbb{R}$, we
use the notation $\mathscr{F}(p,q)\lesssim_{q}\mathscr{G}(p,q)$ to indicate
that for each $q\in\mathscr{Q}$ there exists a constant ${C_{q}\in(0,\infty)}$
such that $\mathscr{F}(p,q)\leq C_{q}\,\mathscr{G}(p,q)$ holds for all
${p\in\mathscr{P}}$. Whenever both relations,
$\mathscr{F}(p,q)\lesssim_{q}\mathscr{G}(p,q)$ and
$\mathscr{G}(p,q)\lesssim_{q}\mathscr{F}(p,q)$, hold simultaneously, we write
$\mathscr{F}(p,q)\eqsim_{q}\mathscr{G}(p,q)$.

For a Banach space $(F,\|\,\cdot\,\|_{F})$ over $\mathbb{R}$, we write
$B_{F}:=\\{x\in F:\|x\|_{F}\leq 1\\}$ for its closed unit ball, and
$\mathcal{B}(F)$ for the Borel $\sigma$-algebra on $(F,\|\,\cdot\,\|_{F})$,
that is the $\sigma$-algebra generated by the open sets. The dual space of all
continuous linear functionals $g\colon F\to\mathbb{R}$ is denoted by
${F}^{\prime}$. We write $g(x)$ or $\langle g,x\rangle$ for the duality
pairing between $g\in{F}^{\prime}$ and $x\in F$, and
$\|g\|_{{F}^{\prime}}:=\sup_{x\in B_{F}}|g(x)|$ for the norm on
${F}^{\prime}$.

Throughout this article, we assume that $(\Omega,\mathcal{A},\mathbb{P})$ is a
complete probability space with expectation operator $\mathbb{E}$, and we mark
statements which hold $\mathbb{P}$-almost surely with $\mathbb{P}$-a.s. For a
second probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ with
expectation operator $\widetilde{\mathbb{E}}$,
$(\Omega\times\widetilde{\Omega},\mathcal{A}\otimes\widetilde{\mathcal{A}},\mathbb{P}\otimes\widetilde{\mathbb{P}})$
denotes the product probability space, i.e., $\Omega\times\widetilde{\Omega}$
is the set of all tuples $(\omega,\widetilde{\omega})$ with $\omega\in\Omega$,
$\widetilde{\omega}\in\widetilde{\Omega}$,
$\mathcal{A}\otimes\widetilde{\mathcal{A}}$ is the product $\sigma$-algebra
generated by all sets of the form $A\times\widetilde{A}$ with
$A\in\mathcal{A}$, $\widetilde{A}\in\widetilde{\mathcal{A}}$, and
$\mathbb{P}\otimes\widetilde{\mathbb{P}}$ is the uniquely defined product
measure satisfying
$(\mathbb{P}\otimes\widetilde{\mathbb{P}})(A\times\widetilde{A})=\mathbb{P}(A)\widetilde{\mathbb{P}}(\widetilde{A})$
for all $A\in\mathcal{A}$ and $\widetilde{A}\in\widetilde{\mathcal{A}}$. The
expectation operator on
$(\Omega\times\widetilde{\Omega},\mathcal{A}\otimes\widetilde{\mathcal{A}},\mathbb{P}\otimes\widetilde{\mathbb{P}})$
will be denoted by $\mathbb{E}\otimes\widetilde{\mathbb{E}}$.

In addition, we let $(E,\|\,\cdot\,\|_{E})$ be a Banach space over
$\mathbb{R}$, the set $\mathbb{N}$ contains all (strictly) positive integers
and, unless otherwise stated, $k\in\mathbb{N}$ is a fixed positive integer
which indicates the order of (statistical) moments.

###  2.2. $k$-fold tensor products of Banach spaces

In this subsection we define (full and symmetric) $k$-fold tensor products of
the Banach space $(E,\|\,\cdot\,\|_{E})$, with the aim of obtaining a new
Banach space satisfying the following:

  1. (i)

It contains the set of all $k$th moments (in Bochner sense) of Bochner
integrable random variables $\xi\colon\Omega\to E$ satisfying
$\mathbb{E}\bigl{[}\|\xi\|_{E}^{k}\bigr{]}<\infty$.

  2. (ii)

The topology on this space (prescribed by its norm) allows to quantify the
convergence of Monte Carlo estimation for statistical moments of order $k$.

For this purpose, symmetry of moments will be particularly important.

We start by defining the (full) _$k$ -fold algebraic tensor product of $E$_,

$\otimes^{k}E=\underbrace{E\otimes\cdots\otimes E}_{\text{$k$ times}},$

that is the vector space consisting of all finite sums of the form

$\sum_{j=1}^{M}x_{j,1}\otimes\cdots\otimes
x_{j,k}=\sum_{j=1}^{M}\bigotimes_{n=1}^{k}x_{j,n},\qquad x_{j,n}\in E,\quad
1\leq j\leq M,\quad 1\leq n\leq k,$

equipped with the algebraic operations rendering the tensor product linear in
each of the $k$ components, see [19, Section 1.1]. The _injective tensor norm_
$\|\,\cdot\,\|_{\varepsilon}$ for an element $U\in\otimes^{k}E$ with
representation $U=\sum_{j=1}^{M}x_{j,1}\otimes\cdots\otimes x_{j,k}$ is
defined by

$\|U\|_{\varepsilon}:=\sup\Biggl{\\{}\biggl{|}\sum_{j=1}^{M}\prod_{n=1}^{k}f_{n}(x_{j,n})\biggr{|}\;\Bigg{|}\;f_{1},\ldots,f_{k}\in
B_{{E}^{\prime}}\Biggr{\\}},$ (2.1)

cf. [54, Section 3.1] and [37, Section 2.3.2]. Note that the value of the
supremum in (2.1) is independent of the choice of the representation of
$U\in\otimes^{k}E$, since

$\biggl{|}\sum_{j=1}^{M}\prod_{n=1}^{k}f_{n}(x_{j,n})\biggr{|}=\biggl{|}\sum_{j=1}^{M}\langle(f_{1}\otimes\cdots\otimes
f_{k}),(x_{j,1}\otimes\cdots\otimes
x_{j,k})\rangle\biggr{|}=|\langle(f_{1}\otimes\cdots\otimes f_{k}),U\rangle|.$

For $k=1$, the Hahn–Banach theorem shows that
$\|\,\cdot\,\|_{\varepsilon}=\|\,\cdot\,\|_{E}$. We call the completion of the
$k$-fold algebraic tensor product space $\otimes^{k}E$ with respect to
$\|\,\cdot\,\|_{\varepsilon}$ in (2.1) the (full) _$k$ -fold injective tensor
product of $E$_ and denote it by $\otimes^{k}_{\varepsilon}E$.

In the context of moments of $E$-valued random variables, we will be
interested in subspaces of $\otimes^{k}E$ and $\otimes^{k}_{\varepsilon}E$
containing only their symmetric elements. To this end, we first introduce for
$x_{1}\otimes\cdots\otimes x_{k}\in\otimes^{k}E$ its _symmetrization_

$s(x_{1}\otimes\cdots\otimes x_{k}):=\frac{1}{k!}\sum_{\sigma\in
S_{k}}x_{\sigma(1)}\otimes\cdots\otimes x_{\sigma(k)},$ (2.2)

where $S_{k}$ is the group of permutations of the set $\\{1,...,k\\}$. The
_$k$ -fold symmetric algebraic tensor product of $E$_, denoted by
$\otimes^{k,s}E$, is then defined as the linear span of
$\\{s(x_{1}\otimes\cdots\otimes x_{k}):x_{1},\ldots,x_{k}\in E\\}$ in
$\otimes^{k}E$, i.e.,

$\displaystyle\otimes^{k,s}E$
$\displaystyle:=\Biggl{\\{}\sum_{j=1}^{M}s(x_{j,1}\otimes\cdots\otimes
x_{j,k})\;\Bigg{|}\;M\in\mathbb{N},\;x_{j,n}\in E,\;1\leq j\leq M,\;1\leq
n\leq k\Biggr{\\}}$
$\displaystyle{\color[rgb]{1,1,1}:}=\Biggl{\\{}\sum_{j=1}^{M}\lambda_{j}\,{\otimes^{k}}x_{j}\;\Bigg{|}\;M\in\mathbb{N},\;\lambda_{j}\in\mathbb{R},\;x_{j}\in
E,\;1\leq j\leq M\Biggr{\\}}$
$\displaystyle{\color[rgb]{1,1,1}:}=\Biggl{\\{}\sum_{j=1}^{M}\delta_{j}\,{\otimes^{k}}x_{j}\;\Bigg{|}\;M\in\mathbb{N},\;\delta_{j}\in\mathscr{E}(k),\;x_{j}\in
E,\;1\leq j\leq M\Biggr{\\}},$

see [19, Section 1.5] or [20, Section 1.1]. Here, we set

${\otimes^{k}}x:=\underbrace{x\otimes\cdots\otimes x}_{\text{$k$
times}}\quad\forall x\in
E,\quad\text{and}\quad\mathscr{E}(k):=\begin{cases}\\{-1,1\\}&\text{if $k$ is
even},\\\ \\{1\\}&\text{if $k$ is odd}.\end{cases}$

The _symmetric injective tensor norm_ $\|\,\cdot\,\|_{\varepsilon_{s}}$ on the
$k$-fold symmetric algebraic tensor product space $\otimes^{k,s}E$ is given by
(see [19, Section 3.1])

$\|U\|_{\varepsilon_{s}}:=\sup\Biggl{\\{}\biggl{|}\sum_{j=1}^{M}\lambda_{j}f(x_{j})^{k}\biggr{|}\;\Bigg{|}\;f\in
B_{{E}^{\prime}}\Biggr{\\}},$ (2.3)

if $U=\sum_{j=1}^{M}\lambda_{j}\,{\otimes^{k}}x_{j}$, where
$\lambda_{j}\in\mathbb{R}$, $x_{j}\in E$ for $1\leq j\leq M$. Note that, as
for the injective tensor norm $\|\,\cdot\,\|_{\varepsilon}$, this definition
does not depend on the choice of the representation of $U$. The _$k$ -fold
symmetric injective tensor product of $E$_, denoted by
$\otimes^{k,s}_{\varepsilon_{s}}E$, is the completion of $\otimes^{k,s}E$ with
respect to the norm $\|\,\cdot\,\|_{\varepsilon_{s}}$ in (2.3).

The symmetrization $s$ in (2.2) extends linearly to a projection
$s\colon{\otimes^{k}}E\to\otimes^{k,s}E$ and, since for
$U=\sum_{j=1}^{M}x_{j,1}\otimes\cdots\otimes x_{j,k}\in\otimes^{k}E$ we
moreover have that

$\begin{split}&\|s(U)\|_{\varepsilon_{s}}\leq\|s(U)\|_{\varepsilon}=\biggl{\|}\sum_{j=1}^{M}\frac{1}{k!}\sum_{\sigma\in
S_{k}}x_{j,\sigma(1)}\otimes\cdots\otimes
x_{j,\sigma(k)}\biggr{\|}_{\varepsilon}\\\
&\quad\leq\frac{1}{k!}\sum_{\sigma\in
S_{k}}\biggl{\|}\sum_{j=1}^{M}x_{j,\sigma(1)}\otimes\cdots\otimes
x_{j,\sigma(k)}\biggr{\|}_{\varepsilon}=\biggl{\|}\sum_{j=1}^{M}x_{j,1}\otimes\cdots\otimes
x_{j,k}\biggr{\|}_{\varepsilon}=\|U\|_{\varepsilon},\end{split}$ (2.4)

it also has a unique continuous extension to a linear projection
$s_{\varepsilon}\colon{\otimes^{k}_{\varepsilon}}E\to\otimes^{k,s}_{\varepsilon_{s}}E$.
Furthermore, on $\otimes^{k,s}_{\varepsilon_{s}}E$ the injective tensor norm
$\|\,\cdot\,\|_{\varepsilon}$ and the symmetric injective tensor norm
$\|\,\cdot\,\|_{\varepsilon_{s}}$ are equivalent, with $k$-dependent
equivalence constants,

$\|U\|_{\varepsilon_{s}}\leq\|U\|_{\varepsilon}\leq\frac{k^{k}}{k!}\|U\|_{\varepsilon_{s}}\quad\forall
U\in\otimes^{k,s}_{\varepsilon_{s}}E.$ (2.5)

see [20, Section 2.3 and 2.7].

######  Remark 2.1.

There are several meaningful options to define norms on the $k$-fold algebraic
tensor product spaces $\otimes^{k}E$ and $\otimes^{k,s}E$. Besides the
injective tensor norm, a common choice is the _projective tensor norm_
$\|\,\cdot\,\|_{\pi}$, defined for $U\in\otimes^{k}E$ by

$\|U\|_{\pi}:=\inf\Biggl{\\{}\sum_{j=1}^{M}\prod_{n=1}^{k}\|x_{j,n}\|_{E}\;\Bigg{|}\;M\in\mathbb{N},\,U=\sum_{j=1}^{M}x_{j,1}\otimes\cdots\otimes
x_{j,k}\Biggr{\\}}.$ (2.6)

The _symmetric projective tensor norm_ on $\otimes^{k,s}E$ is given by (see
[19, Section 2.2])

$\|U\|_{\pi_{s}}:=\inf\Biggl{\\{}\sum_{j=1}^{M}|\lambda_{j}|\,\|x_{j}\|_{E}^{k}\;\Bigg{|}\;M\in\mathbb{N},\,U=\sum_{j=1}^{M}\lambda_{j}\,{\otimes^{k}}x_{j}\Biggr{\\}}.$
(2.7)

Then, for every $U\in\otimes^{k}E$, $\|U\|_{\pi}\geq\|U\|_{\varepsilon}$ and
$\|s(U)\|_{\pi_{s}}\geq\max\\{\|s(U)\|_{\varepsilon_{s}},\|s(U)\|_{\pi}\\}$
hold. The closures of $\otimes^{k}E$ and of $\otimes^{k,s}E$ with respect to
the norms $\|\,\cdot\,\|_{\pi}$ and $\|\,\cdot\,\|_{\pi_{s}}$, respectively,
yield well-defined Banach spaces, the _full_ and _symmetric $k$-fold
projective tensor product of $E$_, denoted by $\otimes^{k}_{\pi}E$ and
$\otimes^{k,s}_{\pi_{s}}E$. As shown in Example 3.21, the projective norm is
not suitable for the error analysis of Monte Carlo methods.

###  2.3. Moments of Banach space valued random variables

The purpose of this subsection is to generalize the notion of the $k$th
moment, defined for a real-valued random variable $X\colon\Omega\to\mathbb{R}$
as

$\mathbb{M}^{k}[X]:=\mathbb{E}\bigl{[}X^{k}\bigr{]}=\int_{\Omega}X(\omega)^{k}\,\mathrm{d}\mathbb{P}(\omega),$

to Banach space valued random variables $\xi\colon\Omega\to E$. To this end,
we first specify the concept of vector-valued integration which we imply when
taking expected values of $E$-valued random variables.

####  2.3.1. Vector-valued integration

An $E$-valued random variable $\xi$ defined on the probability space
$(\Omega,\mathcal{A},\mathbb{P})$ is a mapping $\xi\colon\Omega\to E$ which is
measurable in a certain sense. Specifically, we consider the class of Bochner
measurable random variables; these are all mappings $\xi\colon\Omega\to E$
which are  (i) measurable with respect to the Borel $\sigma$-algebra
$\mathcal{B}(E)$ on $E$, and  (ii) almost surely separably valued, i.e.,
$\xi\in E_{0}$ $\mathbb{P}$-a.s. for some separable subspace $E_{0}\subseteq
E$.  A Bochner measurable random variable $\xi$ is often also called _strongly
measurable_ , emphasizing the contrast to the notion of weak measurability,
which only requires the real-valued random variable $\langle f,\xi\rangle$ to
be measurable for every $f\in{E}^{\prime}$. Note that these notions are
equivalent whenever $\xi$ is almost surely separably valued (e.g., in the case
of a separable Banach space $E$), see [37, Theorem 2.3].

Furthermore, it turns out that $\xi\colon\Omega\to E$ is Bochner measurable if
and only if there exists a sequence of Borel measurable simple functions
$\xi_{n}\colon\Omega\to E$, $n\in\mathbb{N}$, such that $\xi_{n}\to\xi$,
$\mathbb{P}$-a.s. This characterization facilitates the definition of the
Bochner integral

$\int_{\Omega}\xi(\omega)\,\mathrm{d}\mathbb{P}(\omega)=:\mathbb{E}[\xi]\in
E,$

whenever $\xi$ is Bochner measurable and
$\mathbb{E}\bigl{[}\|\xi\|_{E}\bigr{]}<\infty$.

We close this subsection with introducing the corresponding Bochner
$L_{q}$-spaces. For a real Banach space $(F,\|\,\cdot\,\|_{F})$ and
$q\in[1,\infty)$, $L_{q}(\Omega;F):=L_{q}(\Omega,\mathcal{A},\mathbb{P};F)$ is
the space of all (equivalence classes of) $F$-valued Bochner measurable random
variables $\eta\colon\Omega\to F$ such that
$\mathbb{E}\bigl{[}\|\eta\|_{F}^{q}\bigr{]}<\infty$, with norm given by

$\|\eta\|_{L_{q}(\Omega;F)}:=\left(\mathbb{E}\bigl{[}\|\eta\|_{F}^{q}\bigr{]}\right)^{\nicefrac{{1}}{{q}}}=\biggl{(}\int_{\Omega}\|\eta(\omega)\|_{F}^{q}\,\mathrm{d}\mathbb{P}(\omega)\biggr{)}^{\nicefrac{{1}}{{q}}}\\!.$

####  2.3.2. Moments of order $k$

For an $E$-valued random variable $\xi$, its _injective $k$th moment
$\mathbb{M}^{k}_{\varepsilon}[\xi]$_ is defined as the expectation (see e.g.
[37, Section 3.1])

$\mathbb{M}^{k}_{\varepsilon}[\xi]:=\mathbb{E}\bigl{[}\otimes^{k}\xi\bigr{]}=\int_{\Omega}\otimes^{k}\xi(\omega)\,\mathrm{d}\mathbb{P}(\omega)=\int_{\Omega}\underbrace{\xi(\omega)\otimes\cdots\otimes\xi(\omega)}_{\text{$k$
times}}\,\mathrm{d}\mathbb{P}(\omega),$ (2.8)

whenever this integral exists (in Bochner sense) in the $k$-fold injective
tensor product space $\otimes^{k}_{\varepsilon}E$. In what follows, we will
always assume that $\xi\in L_{q}(\Omega;E)$ holds for some $q\in[k,\infty)$.
This guarantees that $\mathbb{M}^{k}_{\varepsilon}[\xi]$ exists in Bochner
sense: Firstly, Bochner measurability of $\xi$ implies that also
$\otimes^{k}\xi\colon\Omega\to\otimes^{k}_{\varepsilon}E$ is Bochner
measurable since the non-linear mapping $E\ni
x\mapsto\otimes^{k}x\in\otimes^{k}_{\varepsilon}E$ is continuous and,
secondly,

$\mathbb{E}\bigl{[}\|{\otimes^{k}}\xi\|_{\varepsilon}\bigr{]}=\mathbb{E}\bigl{[}\|\xi\|_{E}^{k}\bigr{]}\leq\mathbb{E}\bigl{[}\|\xi\|_{E}^{q}\bigr{]}=\|\xi\|_{L_{q}(\Omega;E)}^{q}<\infty.$

Of particular relevance in the present context is that the injective $k$th
moment is an element of the $k$-fold _symmetric_ injective tensor product
space $\otimes^{k,s}_{\varepsilon_{s}}E$, since

$s_{\varepsilon}\bigl{(}\mathbb{M}^{k}_{\varepsilon}[\xi]\bigr{)}=s_{\varepsilon}\bigl{(}\mathbb{E}\bigl{[}\otimes^{k}\xi\bigr{]}\bigr{)}=\mathbb{E}\bigl{[}s_{\varepsilon}\bigl{(}\otimes^{k}\xi\bigr{)}\bigr{]}=\mathbb{E}\bigl{[}\otimes^{k}\xi\bigr{]}=\mathbb{M}^{k}_{\varepsilon}[\xi].$

Here, we have used continuity of the symmetrization
$s_{\varepsilon}\colon{\otimes^{k}_{\varepsilon}}E\to\otimes^{k,s}_{\varepsilon_{s}}E$,
see (2.2) and (2.4), to exchange the order of $s_{\varepsilon}(\,\cdot\,)$ and
$\mathbb{E}[\,\cdot\,]$.

######  Remark 2.2.

The assumption $\xi\in L_{k}(\Omega;E)$ does not only guarantee the existence
of the injective $k$th moment $\mathbb{M}^{k}_{\varepsilon}[\xi]$ but also
that of the projective $k$th moment $\mathbb{M}^{k}_{\pi}[\xi]$, i.e., the
integral in (2.8) converges in Bochner sense also in the stronger projective
tensor norm defined in (2.6), see Remark 2.1. This observation follows from
the identities
$\|{\otimes^{k}}\xi\|_{\pi}=\|\xi\|_{E}^{k}=\|{\otimes^{k}}\xi\|_{\varepsilon}$
showing that the above arguments may be translated verbatim, see also [37,
Theorem 3.8].

##  3\. Monte Carlo estimation of the $k$th moment

This section treats the analysis of abstract standard, single-level and
multilevel Monte Carlo methods to estimate the injective $k$th moment
$\mathbb{M}^{k}_{\varepsilon}[\xi]$ of a Banach space valued random variable
$\xi\in L_{k}(\Omega;E)$. In Subsection 3.1 we first provide necessary
definitions, including those of Rademacher and orthogaussian families as well
as the type of a Banach space. Furthermore, we formulate auxiliary results
based on comparison theorems for Rademacher and Gaussian averages. These
observations facilitate the error analysis for standard and single-level Monte
Carlo estimation in Subsection 3.2 and for the multilevel Monte Carlo method
in Subsection 3.3.

###  3.1. Auxiliary results on Rademacher and Gaussian averages

######  Definition 3.1 (Rademacher family).

Let $(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ be a
complete probability space and $r_{j}\colon\widetilde{\Omega}\to\\{-1,1\\}$,
$j\in J\subseteq\mathbb{N}$, be a family of independent random variables such
that
$\widetilde{\mathbb{P}}(r_{j}=-1)=\widetilde{\mathbb{P}}(r_{j}=1)=\tfrac{1}{2}$
for all $j\in J$. Then, the collection of random variables $(r_{j})_{j\in J}$
is called a _Rademacher family_.

######  Definition 3.2 (Orthogaussian family).

Let $(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ be a
complete probability space with expectation $\widetilde{\mathbb{E}}$. Suppose
that $g_{j}\colon\widetilde{\Omega}\to\mathbb{R}$, $j\in
J\subseteq\mathbb{N}$, are independent standard Gaussian random variables,
i.e., $\widetilde{\mathbb{E}}[g_{j}]=0$,
$\widetilde{\mathbb{E}}\bigl{[}g_{j}^{2}\bigr{]}=1$ for all $j\in J$, and
$\widetilde{\mathbb{E}}[g_{i}g_{j}]=0$ for $i\neq j$. Then, the collection
$(g_{j})_{j\in J}$ is called an _orthogaussian family_.

Assuming that $(z_{j})_{j=1}^{M}$ is a Rademacher or orthogaussian family and
$(x_{j})_{j=1}^{M}$ are vectors in the Banach space $E$, the $E$-norm of the
finite random sum $\sum_{j=1}^{M}z_{j}x_{j}$ is the supremum of a real-valued
(Rademacher or Gaussian) stochastic process. More specifically, the
Hahn–Banach theorem allows us to rewrite the norm as follows,

$\biggl{\|}\sum_{j=1}^{M}z_{j}x_{j}\biggr{\|}_{E}=\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}z_{j}f(x_{j})\biggr{|}=\sup_{(t_{1},\ldots,t_{M})\in
T}\biggl{|}\sum_{j=1}^{M}z_{j}t_{j}\biggr{|},$ (3.1)

where $T$ is the compact subset of $\mathbb{R}^{M}$ given by
$T:=\\{(f(x_{1}),\ldots,f(x_{M})):f\in B_{{E}^{\prime}}\\}$.

The next lemma summarizes comparison theorems for Gaussian and Rademacher
averages, see [47, Corollary 3.17 & Theorem 4.12]. It will facilitate
generalizing the equality (3.1) to an upper bound for $L_{q}$-norms of random
variables of the form

$\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}r_{j}f(x_{j})^{k_{j}}\biggr{|}\quad\text{and}\quad\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}f(x_{j})^{k_{j}}\biggr{|},\qquad
k_{j}\in\mathbb{N},\quad 1\leq j\leq M.$

######  Lemma 3.3.

Let $M\in\mathbb{N}$, and let $z:=(z_{j})_{j=1}^{M}$ be a Rademacher or
orthogaussian family on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$. Suppose
that $(\varphi_{j})_{j=1}^{M}$ are functions on $\mathbb{R}$ such that, for
every $1\leq j\leq M$,

$\varphi_{j}(0)=0\qquad\text{and}\qquad|\varphi_{j}(s)-\varphi_{j}(t)|\leq|s-t|\quad\forall
s,t\in\mathbb{R}.$ (3.2)

Assume further that $G\colon[0,\infty)\to[0,\infty)$ is convex and increasing.
Then, we have for any bounded subset $T\subset\mathbb{R}^{M}$ the relation

$\widetilde{\mathbb{E}}G\Biggl{(}\frac{1}{2}\sup_{(t_{1},\ldots,t_{M})\in
T}\biggl{|}\sum_{j=1}^{M}z_{j}\varphi_{j}(t_{j})\biggr{|}\Biggr{)}\leq\widetilde{\mathbb{E}}G\Biggl{(}C_{z}\sup_{(t_{1},\ldots,t_{M})\in
T}\biggl{|}\sum_{j=1}^{M}z_{j}t_{j}\biggr{|}\Biggr{)},$ (3.3)

where $\widetilde{\mathbb{E}}$ denotes the expectation operator on
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$, and
$C_{z}$ is given by

$C_{z}:=\begin{cases}1&\text{ if $z$ is a Rademacher family},\\\ 2&\text{ if
$z$ is an orthogaussian family}.\end{cases}$ (3.4)

######  Proposition 3.4.

Let $M\in\mathbb{N}$, $x_{1},\ldots,x_{M}\in E$, and $(z_{j})_{j=1}^{M}$ be a
Rademacher or orthogaussian family on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ (with
expectation $\widetilde{\mathbb{E}}$). In addition, assume that
$G\colon[0,\infty)\to[0,\infty)$ is a convex and increasing function.

  1. (i)

For integers $k_{1},\ldots,k_{M}\in\mathbb{N}$, we have the relation

$\widetilde{\mathbb{E}}G\Biggl{(}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}z_{j}f(x_{j})^{k_{j}}\biggr{|}\Biggr{)}\leq\widetilde{\mathbb{E}}G\Biggl{(}2C_{z}\,\biggl{\|}\sum_{j=1}^{M}z_{j}k_{j}\|x_{j}\|_{E}^{k_{j}-1}x_{j}\biggr{\|}_{E}\Biggr{)}.$
(3.5)

  2. (ii)

For general exponents $q_{1},\ldots,q_{M}\in[1,\infty)$, the following holds:

$\widetilde{\mathbb{E}}G\Biggl{(}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}z_{j}|f(x_{j})|^{q_{j}}\biggr{|}\Biggr{)}\leq\widetilde{\mathbb{E}}G\Biggl{(}2C_{z}\,\biggl{\|}\sum_{j=1}^{M}z_{j}q_{j}\|x_{j}\|_{E}^{q_{j}-1}x_{j}\biggr{\|}_{E}\Biggr{)}.$
(3.6)

Here, the constant $C_{z}\in\\{1,2\\}$ is defined as in (3.4).

######  Remark 3.5.

Part (i) of Proposition 3.4 is a generalization of the observation made by
Ledoux and Talagrand in [47, Equation (4.19)]; there formulated for Rademacher
averages, $G(x):=x$, and $k_{1}=\ldots=k_{M}=2$. That is, we generalize to
higher-order polynomials on one hand, and to Gaussian averages on the other
hand.

###### Proof of Proposition 3.4.

We will prove (i) and (ii) using Lemma 3.3. Without loss of generality we may
assume that $x_{j}\neq 0$ for all $1\leq j\leq M$.

To derive (i), let $M\in\mathbb{N}$ and $k_{j}\in\mathbb{N}$,
$\widetilde{x}_{j}\in E\setminus\\{0\\}$ for all $1\leq j\leq M$. Furthermore,
for $1\leq j\leq M$, define
$I_{j}:=\bigl{[}-\|\widetilde{x}_{j}\|_{E},\|\widetilde{x}_{j}\|_{E}\bigr{]}$
and

$\widetilde{\varphi}_{j}\colon
I_{j}\to\mathbb{R},\qquad\widetilde{\varphi}_{j}(s):=k_{j}^{-1}\|\widetilde{x}_{j}\|_{E}^{1-k_{j}}s^{k_{j}},\quad
s\in I_{j}.$ (3.7)

The function $\widetilde{\varphi}_{j}$ is continuously differentiable on the
interior of $I_{j}$ with $|\widetilde{\varphi}_{j}^{\prime}(s)|\leq 1$. Let
$P_{j}\colon\mathbb{R}\to I_{j}$ denote the projection
$P_{j}(s):=\max\\{-\|\widetilde{x}_{j}\|_{E},\min\\{s,\|\widetilde{x}_{j}\|_{E}\\}\\}$
onto the interval $I_{j}$. Then, for every $1\leq j\leq M$, the function

$\varphi_{j}\colon\mathbb{R}\to\mathbb{R},\qquad\varphi_{j}(s):=\widetilde{\varphi}_{j}(P_{j}(s)),$

satisfies the assumptions (3.2) of Lemma 3.3. Therefore, letting the bounded
set $T\subset\mathbb{R}^{M}$ be given by
$T:=\\{(f(\widetilde{x}_{1}),\ldots,f(\widetilde{x}_{M})):f\in
B_{{E}^{\prime}}\\}$, we may combine the observation (3.1) with (3.3) and by
also noting that $T\subseteq I_{1}\times\ldots\times I_{M}$ we obtain that,
for every convex and increasing function $G\colon[0,\infty)\to[0,\infty)$,

$\displaystyle\widetilde{\mathbb{E}}G\Biggl{(}\frac{1}{2}\sup_{f\in
B_{{E}^{\prime}}}$
$\displaystyle\biggl{|}\sum_{j=1}^{M}z_{j}k_{j}^{-1}\|\widetilde{x}_{j}\|_{E}^{1-k_{j}}f(\widetilde{x}_{j})^{k_{j}}\biggr{|}\Biggr{)}=\widetilde{\mathbb{E}}G\Biggl{(}\frac{1}{2}\sup_{(t_{1},\ldots,t_{M})\in
T}\biggl{|}\sum_{j=1}^{M}z_{j}\varphi_{j}(t_{j})\biggr{|}\Biggr{)}$
$\displaystyle\quad\leq\widetilde{\mathbb{E}}G\Biggl{(}C_{z}\sup_{(t_{1},\ldots,t_{M})\in
T}\biggl{|}\sum_{j=1}^{M}z_{j}t_{j}\biggr{|}\Biggr{)}=\widetilde{\mathbb{E}}G\Biggl{(}C_{z}\,\biggl{\|}\sum_{j=1}^{M}z_{j}\widetilde{x}_{j}\biggr{\|}_{E}\Biggr{)}.$
(3.8)

Finally, for $x_{1},\ldots,x_{M}\in E\setminus\\{0\\}$, we choose
$\widetilde{x}_{j}:=2k_{j}\|x_{j}\|_{E}^{k_{j}-1}x_{j}\in E\setminus\\{0\\}$
for all $1\leq j\leq M$ and (3.8) shows (3.5).

For (ii) we modify the above arguments by replacing $k_{j}$ and the interval
$I_{j}$ in the definition (3.7) of $\widetilde{\varphi}_{j}$ with $q_{j}$ and
$I_{j}:=\bigl{[}0,\|\widetilde{x}_{j}\|_{E}\bigr{]}$, respectively, and the
projection $P_{j}\colon\mathbb{R}\to I_{j}$ with
$P_{j}(s):=\min\\{|s|,\|\widetilde{x}_{j}\|_{E}\\}$. Then, the function
$\varphi_{j}(s):=\widetilde{\varphi}_{j}(P_{j}(s))$ satisfies the assumptions
(3.2) of Lemma 3.3, since $\varphi_{j}(0)=0$,
$|\widetilde{\varphi}_{j}^{\prime}(s)|\leq 1$ for all
$s\in(0,\|\widetilde{x}_{j}\|_{E})$ and by the mean value theorem combined
with the reverse triangle inequality we thus have that

$|\varphi_{j}(s)-\varphi_{j}(t)|=|\widetilde{\varphi}_{j}(P_{j}(s))-\widetilde{\varphi}_{j}(P_{j}(t))|\leq|P_{j}(s)-P_{j}(t)|\leq\bigl{|}|s|-|t|\bigr{|}\leq|s-t|.$

Furthermore, for every $f\in B_{{E}^{\prime}}$, we have that
$P_{j}(f(\widetilde{x}_{j}))=|f(\widetilde{x}_{j})|$ and we obtain the
analogue of (3.8),

$\widetilde{\mathbb{E}}G\Biggl{(}\frac{1}{2}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}z_{j}q_{j}^{-1}\|\widetilde{x}_{j}\|_{E}^{1-q_{j}}|f(\widetilde{x}_{j})|^{q_{j}}\biggr{|}\Biggr{)}\leq\widetilde{\mathbb{E}}G\Biggl{(}C_{z}\,\biggl{\|}\sum_{j=1}^{M}z_{j}\widetilde{x}_{j}\biggr{\|}_{E}\Biggr{)}.$

The choice $\widetilde{x}_{j}:=2q_{j}\|x_{j}\|_{E}^{q_{j}-1}x_{j}$, $1\leq
j\leq M$, completes the proof of (3.6). ∎

The next lemma shows that we may symmetrize independent random variables with
vanishing mean, when bounding $L_{q}$-norms of their sum. This result can be
found, e.g., in [47, Lemma 6.3] or [14, Lemma 5.9].

######  Lemma 3.6 (Symmetrization).

Let $q\in[1,\infty)$, $M\in\mathbb{N}$, $(r_{j})_{j=1}^{M}$ be a Rademacher
family on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$, and let
$\eta_{1},\ldots,\eta_{M}\in L_{q}(\widetilde{\Omega};F)$ be centered random
variables, i.e., $\widetilde{\mathbb{E}}[\eta_{j}]=0$ for $1\leq j\leq M$,
taking values in a real Banach space $(F,\|\,\cdot\,\|_{F})$ such that
$\eta_{1},\ldots,\eta_{M},r_{1},\ldots,r_{M}$ are independent. Then,

$\biggl{\|}\sum_{j=1}^{M}\eta_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};F)}\leq
2\,\biggl{\|}\sum_{j=1}^{M}r_{j}\eta_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};F)}.$

######  Definition 3.7 (Kahane–Khintchine constants).

Assume that $p,q\in[1,\infty)$. The $(q,p)$ Kahane–Khintchine constant
$K_{q,p}$ is the smallest constant $K\in(0,\infty)$ such that for any
Rademacher family $(r_{j})_{j\in\mathbb{N}}$ on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$, for any
real Banach space $(F,\|\,\cdot\,\|_{F})$, for all $n\in\mathbb{N}$, and every
$x_{1},\ldots,x_{n}\in F$,

$\biggl{\|}\sum_{j=1}^{n}r_{j}x_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};F)}\leq
K\,\biggl{\|}\sum_{j=1}^{n}r_{j}x_{j}\biggr{\|}_{L_{p}(\widetilde{\Omega};F)}.$
(3.9)

######  Remark 3.8.

For the case $q\leq p$, Hölder’s inequality shows that ${K_{q,p}=1}$.
Finiteness of the constant $K_{q,p}$ in the non-trivial case $q>p$ was derived
by Kahane [39]; it implies that, for Rademacher sums, all $L_{q}$-norms with
$q\in[1,\infty)$ are equivalent.

######  Remark 3.9.

By invoking an argument based on the central limit theorem (see [47, p. 103])
the Kahane–Khintchine inequality for Rademacher sums (3.9) implies a
corresponding result for Gaussian averages: For all $p,q\in[1,\infty)$, for
any orthogaussian family $(g_{j})_{j\in\mathbb{N}}$ on a complete probability
space $(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$,
for any real Banach space $(F,\|\,\cdot\,\|_{F})$, for all $n\in\mathbb{N}$,
and every $x_{1},\ldots,x_{n}\in F$, we have that

$\biggl{\|}\sum_{j=1}^{n}g_{j}x_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};F)}\leq
K_{q,p}\,\biggl{\|}\sum_{j=1}^{n}g_{j}x_{j}\biggr{\|}_{L_{p}(\widetilde{\Omega};F)}.$
(3.10)

######  Definition 3.10 (Type $p$ constant).

A real Banach space $(F,\|\,\cdot\,\|_{F})$ is said to be of _(Rademacher)
type_ $p\in[1,2]$ if there exists a constant $\tau\in(0,\infty)$ such that for
any Rademacher family $(r_{j})_{j\in\mathbb{N}}$ on a complete probability
space $(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$
(with expectation $\widetilde{\mathbb{E}}$), for every $n\in\mathbb{N}$, and
all vectors $x_{1},\ldots,x_{n}\in F$,

$\biggl{\|}\sum_{j=1}^{n}r_{j}x_{j}\biggr{\|}_{L_{p}(\widetilde{\Omega};F)}=\Biggl{(}\widetilde{\mathbb{E}}\Biggl{[}\biggl{\|}\sum_{j=1}^{n}r_{j}x_{j}\biggr{\|}_{F}^{p}\Biggr{]}\Biggr{)}^{\nicefrac{{1}}{{p}}}\leq\tau\Biggl{(}\sum_{j=1}^{n}\|x_{j}\|_{F}^{p}\Biggr{)}^{\nicefrac{{1}}{{p}}}\\!.$
(3.11)

In this case, the smallest constant $\tau\in(0,\infty)$ in (3.11) is called
the _type $p$ constant of $F$_ and will be denoted by $\tau_{p}(F)$.

######  Remark 3.11.

The definition of the type of a Banach space $(F,\|\,\cdot\,\|_{F})$ is often
complemented with the notion of its cotype: $F$ has _cotype_ $q\in[2,\infty]$
if there exists a constant $c\in(0,\infty)$ such that for any Rademacher
family $(r_{j})_{j\in\mathbb{N}}$ on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$, for
every $n\in\mathbb{N}$, and all vectors $x_{1},\ldots,x_{n}\in F$,

$\displaystyle\qquad\quad\Biggl{(}\sum_{j=1}^{n}\|x_{j}\|_{F}^{q}\Biggr{)}^{\nicefrac{{1}}{{q}}}$
$\displaystyle\leq
c\,\biggl{\|}\sum_{j=1}^{n}r_{j}x_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};F)}$
$\displaystyle\text{if }q\in[2,\infty),\qquad\quad$ $\displaystyle\sup_{1\leq
j\leq n}\|x_{j}\|_{F}$ $\displaystyle\leq
c\,\biggl{\|}\sum_{j=1}^{n}r_{j}x_{j}\biggr{\|}_{L_{1}(\widetilde{\Omega};F)}$
$\displaystyle\text{if }q=\infty.$

######  Remark 3.12.

Every Banach space has type $1$ and cotype $\infty$ by the triangle inequality
and Lévy’s inequality (see e.g. [47, Proposition 2.3]), respectively.
Conversely, by the (classical) Khintchine inequalities (see e.g. [47, Lemma
4.1]) there exist constants $A_{q},B_{q}\in(0,\infty)$ depending only on
$q\in[1,\infty)$ such that, for any Rademacher family
$(r_{j})_{j\in\mathbb{N}}$ on
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ and any
finite sequence $(a_{j})_{j=1}^{n}$ of real numbers,

$A_{q}\Biggl{(}\sum_{j=1}^{n}a_{j}^{2}\Biggr{)}^{\nicefrac{{1}}{{2}}}\leq\Biggl{(}\widetilde{\mathbb{E}}\Biggl{[}\biggl{|}\sum_{j=1}^{n}r_{j}a_{j}\biggr{|}^{q}\Biggr{]}\Biggr{)}^{\nicefrac{{1}}{{q}}}=\biggl{\|}\sum_{j=1}^{n}r_{j}a_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\leq
B_{q}\Biggl{(}\sum_{j=1}^{n}a_{j}^{2}\Biggr{)}^{\nicefrac{{1}}{{2}}}$ (3.12)

which implies that the type cannot be larger than $2$ and the cotype cannot be
smaller than $2$. Kwapień [45] showed that a Banach space has type $2$ and
cotype $2$ if and only if it is isomorphic to a Hilbert space.

######  Example 3.13.

Let $(H,(\,\cdot\,,\,\cdot\,)_{H})$ be a real separable Hilbert space. In this
case, the _Hilbert tensor product space_ $\otimes_{w_{2}}^{2}H$ (see Appendix
A for the definition) is again a Hilbert space and, consequently, has type
$p=2$. However, none of the tensor product spaces $\otimes_{\pi}^{2}H$,
$\otimes^{2,s}_{\pi_{s}}H$, $\otimes_{\varepsilon}^{2}H$ or
$\otimes^{2,s}_{\varepsilon_{s}}H$ has a non-trivial type ${p>1}$.

This can be seen by the following counterexample: Let
$(e_{j})_{j\in\mathbb{N}}$ be an orthonormal basis for $H$. Then, for all
$p\in[1,\infty)$ and every $n\in\mathbb{N}$, we have

$\Biggl{(}\sum_{j=1}^{n}\|e_{j}\otimes
e_{j}\|_{\pi}^{p}\Biggr{)}^{\nicefrac{{1}}{{p}}}=\Biggl{(}\sum_{j=1}^{n}\|e_{j}\otimes
e_{j}\|_{\varepsilon}^{p}\Biggr{)}^{\nicefrac{{1}}{{p}}}=\Biggl{(}\sum_{j=1}^{n}\|e_{j}\|_{H}^{2p}\Biggr{)}^{\nicefrac{{1}}{{p}}}=n^{\nicefrac{{1}}{{p}}}.$

Moreover, the calculations in Appendix A (see the identities (A.2) and (A.3)
of Lemma A.1) imply that for any Rademacher family $(r_{j})_{j\in\mathbb{N}}$
on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$, for all
$p\in[1,\infty)$ and for every $n\in\mathbb{N}$,

$\displaystyle\biggl{\|}\sum_{j=1}^{n}r_{j}\,e_{j}\otimes
e_{j}\biggr{\|}_{L_{p}(\widetilde{\Omega};\otimes^{2}_{\pi}H)}=n\qquad\text{and}\qquad\biggl{\|}\sum_{j=1}^{n}r_{j}\,e_{j}\otimes
e_{j}\biggr{\|}_{L_{p}(\widetilde{\Omega};\otimes^{2}_{\varepsilon}H)}=1,$

and the same statements hold when replacing $\otimes^{2}_{\pi}H$ by the
symmetric projective tensor product space $\otimes^{2,s}_{\pi_{s}}H$ and
$\otimes^{2}_{\varepsilon}H$ by the symmetric injective tensor product space
$\otimes^{2,s}_{\varepsilon_{s}}H$, respectively. This shows that (i) neither
$\otimes^{2}_{\pi}H$ nor $\otimes^{2,s}_{\pi_{s}}H$ have a non-trivial type
$p>1$, and (ii) neither $\otimes^{2}_{\varepsilon}H$ nor
$\otimes^{2,s}_{\varepsilon_{s}}H$ have a non-trivial cotype $q<\infty$. Thus,
$\otimes^{2}_{\varepsilon}H$ and $\otimes^{2,s}_{\varepsilon_{s}}H$ do not
have a non-trivial type either, cf. [36, Theorem 7.1.14].

In the next subsections we will see that the type $p\in[1,2]$ of a Banach
space $E$ determines the rate of convergence when approximating statistical
moments of $E$-valued random variables by means of Monte Carlo methods and,
moreover, that this convergence rate does not depend on the order $k$ of the
moment. However, as the above example illustrates, to derive this finding, it
is not possible to argue by transferring the type of a Banach space to its
$k$-fold tensor product.

###  3.2. Standard and single-level Monte Carlo estimation

The next proposition is the key result for proving convergence of Monte Carlo
methods for means, i.e., statistical moments of order $k=1$. It can be found,
e.g., in [47, Proposition 9.11] for the case $q=p$ and in this generality in
[14, Proposition 5.10].

######  Proposition 3.14.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in[1,2]$. Let
$q\in[p,\infty)$, $M\in\mathbb{N}$ and $\eta_{1},\ldots,\eta_{M}\in
L_{q}(\Omega;E)$ be independent $E$-valued random variables with vanishing
mean, $\mathbb{E}[\eta_{j}]=0$ for all $1\leq j\leq M$. Then,

$\biggl{\|}\sum_{j=1}^{M}\eta_{j}\biggr{\|}_{L_{q}(\Omega;E)}\leq
2K_{q,p}\tau_{p}(E)\Biggl{(}\sum_{j=1}^{M}\|\eta_{j}\|_{L_{q}(\Omega;E)}^{p}\Biggr{)}^{\nicefrac{{1}}{{p}}}\\!.$

######  Corollary 3.15.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in[1,2]$ and let
$\eta\in L_{1}(\Omega;E)$. In addition, let $q\in[p,\infty)$, $M\in\mathbb{N}$
and $\xi_{1},\ldots,\xi_{M}\in L_{q}(\Omega;E)$ be independent and identically
distributed $E$-valued random variables. Then,

$\displaystyle\biggl{\|}\mathbb{E}[\eta]-\frac{1}{M}\sum_{j=1}^{M}\xi_{j}\biggr{\|}_{L_{q}(\Omega;E)}$
$\displaystyle\leq\bigl{\|}\mathbb{E}[\eta-\xi_{1}]\bigr{\|}_{E}$
$\displaystyle\quad+2K_{q,p}\tau_{p}(E)\,M^{-\left(1-\frac{1}{p}\right)}\bigl{\|}\xi_{1}-\mathbb{E}[\xi_{1}]\bigr{\|}_{L_{q}(\Omega;E)}.$

###### Proof.

By applying the triangle inequality on $L_{q}(\Omega;E)$ and Proposition 3.14
(noting that $\xi_{j}-\mathbb{E}[\xi_{1}]$, $1\leq j\leq M$, are independent
and centered), we find that

$\displaystyle\biggl{\|}\mathbb{E}[\eta]-\frac{1}{M}\sum_{j=1}^{M}\xi_{j}$
$\displaystyle\biggr{\|}_{L_{q}(\Omega;E)}\leq\bigl{\|}\mathbb{E}[\eta-\xi_{1}]\bigr{\|}_{E}+\frac{1}{M}\,\biggl{\|}\sum_{j=1}^{M}\bigl{(}\xi_{j}-\mathbb{E}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega;E)}$
$\displaystyle\leq\bigl{\|}\mathbb{E}[\eta-\xi_{1}]\bigr{\|}_{E}+2K_{q,p}\tau_{p}(E)\,M^{-1}\Biggl{(}\sum_{j=1}^{M}\bigl{\|}\xi_{j}-\mathbb{E}[\xi_{1}]\bigr{\|}_{L_{q}(\Omega;E)}^{p}\Biggr{)}^{\nicefrac{{1}}{{p}}}\\!,$

and the claim follows by the identical distribution of
$\xi_{1},\ldots,\xi_{M}$. ∎

The remainder of this subsection is devoted to generalizing the approximation
result of Monte Carlo estimation for the first statistical moment in Corollary
3.15 to (injective) statistical moments $\mathbb{M}^{k}_{\varepsilon}[\eta]$
of an arbitrary order $k\in\mathbb{N}$. Example 3.13 shows that it is in
general not possible to argue via the type of the tensor product space. We
therefore prove the convergence rates of Monte Carlo methods directly by means
of the auxiliary results derived in Subsection 3.1.

######  Theorem 3.16.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in[1,2]$. Let
$q\in[p,\infty)$, $k,M\in\mathbb{N}$ and $\xi_{1},\ldots,\xi_{M}\in
L_{kq}(\Omega;E)$ be independent and identically distributed $E$-valued random
variables. Then,

$\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{k}\xi_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}\leq
C^{\,\sf
SL}_{q,p,k}\,M^{-\left(1-\frac{1}{p}\right)}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k}.$
(3.13)

Here, we recall the constant $B_{q}\in(0,\infty)$ from the classical
Khintchine inequalities (3.12), as well as the Kahane–Khintchine constant
$K_{q,p}$ and type $p$ constant $\tau_{p}(E)$ from Definitions 3.7 and 3.10,
respectively, and set

$C^{\,\sf SL}_{q,p,k}:=2(2kK_{q,p}\tau_{p}(E)+B_{q}).$ (3.14)

###### Proof.

Assume that $(r_{j})_{j=1}^{M}$ is a Rademacher family on a complete
probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ and, for
$j\in\\{1,\ldots,M\\}$, let
$\boldsymbol{\xi}_{j}\colon\Omega\times\widetilde{\Omega}\to E$ and
$\boldsymbol{r}_{j}\colon\Omega\times\widetilde{\Omega}\to\\{-1,1\\}$ denote
the mappings that satisfy
$\boldsymbol{\xi}_{j}(\omega,\widetilde{\omega})=\xi_{j}(\omega)$ and
$\boldsymbol{r}_{j}(\omega,\widetilde{\omega})=r_{j}(\widetilde{\omega})$ for
all $(\omega,\widetilde{\omega})\in\Omega\times\widetilde{\Omega}$. Notice
that on
$(\Omega\times\widetilde{\Omega},\mathcal{A}\otimes\widetilde{\mathcal{A}},\mathbb{P}\otimes\widetilde{\mathbb{P}})$
the random variables $(\boldsymbol{r}_{j})_{j=1}^{M}$ form a Rademacher family
and
$\boldsymbol{\xi}_{1},\ldots,\boldsymbol{\xi}_{M},\boldsymbol{r}_{1},\ldots,\boldsymbol{r}_{M}$
are independent. Moreover, $\xi_{j}\in L_{kq}(\Omega;E)$ implies that
$\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\in\otimes^{k,s}_{\varepsilon_{s}}E$ and
$\otimes^{k}\boldsymbol{\xi}_{j}\in
L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)$ are
well-defined. We further note that by the identical distribution of
$\xi_{1},\ldots,\xi_{M}$,

$(\mathbb{E}\otimes\widetilde{\mathbb{E}})\bigl{[}\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{]}=\mathbb{E}\bigl{[}\otimes^{k}\xi_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{]}=\mathbb{E}\bigl{[}\otimes^{k}\xi_{j}\bigr{]}-\mathbb{M}^{k}_{\varepsilon}[\xi_{j}]=0.$

This shows that the independent random variables
$\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\colon\Omega\times\widetilde{\Omega}\to\otimes^{k,s}_{\varepsilon_{s}}E$,
$1\leq j\leq M$, are centered. Therefore, we can use Lemma 3.6 on the
probability space
$(\Omega\times\widetilde{\Omega},\mathcal{A}\otimes\widetilde{\mathcal{A}},\mathbb{P}\otimes\widetilde{\mathbb{P}})$
and for the Banach space $\otimes^{k,s}_{\varepsilon_{s}}E$ to deduce that

$\biggl{\|}\sum_{j=1}^{M}\bigl{(}\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\leq
2\,\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\bigl{(}\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}.$

By the triangle inequality on
$L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)$ we
then obtain that

$\begin{split}\biggl{\|}\sum_{j=1}^{M}&\bigl{(}\otimes^{k}\xi_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}=\biggl{\|}\sum_{j=1}^{M}\bigl{(}\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&\leq
2\,\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\bigl{(}\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&\leq
2\,\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\otimes^{k}\\!\boldsymbol{\xi}_{j}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}+2\,\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&=:2\text{(A)}+2\text{(B)}.\end{split}$ (3.15)

To bound term (A) from above, we apply Fubini’s theorem and the
Kahane–Khintchine inequality (3.9) for the Banach space
$F:=\otimes^{k,s}_{\varepsilon_{s}}E$ and obtain that

(A)
$\displaystyle=\Biggl{(}\int_{\Omega}\biggl{\|}\sum_{j=1}^{M}r_{j}(\,\cdot\,)\otimes^{k}\\!\xi_{j}(\omega)\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}^{q}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}$
$\displaystyle\leq
K_{q,p}\Biggl{(}\int_{\Omega}\biggl{\|}\sum_{j=1}^{M}r_{j}(\,\cdot\,)\otimes^{k}\\!\xi_{j}(\omega)\biggr{\|}_{L_{p}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}^{q}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}\\!.$

Upon inserting the definition (2.3) of the symmetric injective tensor norm, we
use Proposition 3.4(i) for the convex increasing function $G(t):=t^{p}$,
$t\geq 0$, and the fact that the Banach space $E$ has type $p\in[1,2]$ with
type constant $\tau_{p}(E)\in(0,\infty)$, to conclude that

(A) $\displaystyle\leq
K_{q,p}\Biggl{(}\int_{\Omega}\Biggl{(}\widetilde{\mathbb{E}}\Biggl{[}\biggl{(}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}r_{j}(\,\cdot\,)f\bigl{(}\xi_{j}(\omega)\bigr{)}^{k}\biggr{|}\biggr{)}^{p}\Biggr{]}\Biggr{)}^{\nicefrac{{q}}{{p}}}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}$
$\displaystyle\leq
2kK_{q,p}\Biggl{(}\int_{\Omega}\Biggl{(}\widetilde{\mathbb{E}}\Biggl{[}\biggl{\|}\sum_{j=1}^{M}r_{j}(\,\cdot\,)\|\xi_{j}(\omega)\|_{E}^{k-1}\xi_{j}(\omega)\biggr{\|}_{E}^{p}\Biggr{]}\Biggr{)}^{\nicefrac{{q}}{{p}}}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}$
(A) $\displaystyle\leq
2kK_{q,p}\tau_{p}(E)\Biggl{(}\int_{\Omega}\Biggl{(}\sum_{j=1}^{M}\|\xi_{j}(\omega)\|_{E}^{kp}\Biggr{)}^{\nicefrac{{q}}{{p}}}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}$
$\displaystyle=2kK_{q,p}\tau_{p}(E)\biggl{\|}\sum_{j=1}^{M}\|\xi_{j}\|_{E}^{kp}\biggr{\|}_{L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})}^{\nicefrac{{1}}{{p}}}.$

Since $q\geq p$, we can use the triangle inequality on
$L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})$, yielding

$\text{(A)}\leq
2kK_{q,p}\tau_{p}(E)\Biggl{(}\sum_{j=1}^{M}\|\xi_{j}\|_{L_{kq}(\Omega;E)}^{kp}\biggr{)}^{\nicefrac{{1}}{{p}}}=2kK_{q,p}\tau_{p}(E)M^{\nicefrac{{1}}{{p}}}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k},$
(3.16)

where we also used the identical distribution of $\xi_{1},\ldots,\xi_{M}$.

For term (B) we use the estimate

$\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}=\bigl{\|}\mathbb{E}\bigl{[}\otimes^{k}\xi_{1}\bigr{]}\bigr{\|}_{\varepsilon_{s}}\leq\mathbb{E}\bigl{[}\|{\otimes^{k}}\xi_{1}\|_{\varepsilon_{s}}\bigr{]}=\mathbb{E}\bigl{[}\|\xi_{1}\|_{E}^{k}\bigr{]}\leq\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k},$

as well as the classical Khintchine inequalities (3.12) so that

$\biggl{\|}\sum_{j=1}^{M}r_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\leq
B_{q}\,M^{\nicefrac{{1}}{{2}}}\leq B_{q}\,M^{\nicefrac{{1}}{{p}}}\\!,$

and conclude that

$\begin{split}\text{(B)}&=\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\sum_{j=1}^{M}\boldsymbol{r}_{j}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}=\Biggl{(}\int_{\widetilde{\Omega}}\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\sum_{j=1}^{M}r_{j}(\widetilde{\omega})\biggr{\|}_{\varepsilon_{s}}^{q}\,\mathrm{d}\widetilde{\mathbb{P}}(\widetilde{\omega})\Biggr{)}^{\nicefrac{{1}}{{q}}}\\\
&=\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}\Biggl{(}\int_{\widetilde{\Omega}}\biggl{|}\sum_{j=1}^{M}r_{j}(\widetilde{\omega})\biggr{|}^{q}\,\mathrm{d}\widetilde{\mathbb{P}}(\widetilde{\omega})\Biggr{)}^{\nicefrac{{1}}{{q}}}\leq
B_{q}\,M^{\nicefrac{{1}}{{p}}}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k}.\end{split}$
(3.17)

Finally, combining (3.15), (3.16) and (3.17) shows that

$\displaystyle\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{k}\xi_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle=M^{-1}\,\biggl{\|}\sum_{j=1}^{M}\bigl{(}\otimes^{k}\xi_{j}-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle\leq
2(2kK_{q,p}\tau_{p}(E)+B_{q})\,M^{-\left(1-\frac{1}{p}\right)}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k},$

which along with the definition (3.14) of $C^{\,\sf SL}_{q,p,k}$ completes the
proof of (3.13). ∎

The estimate (3.16) of term (A) in the proof of Theorem 3.16 reveals the
following analogue of Proposition 3.14 for independent (not necessarily
identically distributed) random variables $\eta_{1},\ldots,\eta_{M}\in
L_{kq}(\Omega;E)$ with vanishing $k$th moment.

######  Corollary 3.17.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in[1,2]$. Let
$q\in[p,\infty)$, $k,M\in\mathbb{N}$ and $\eta_{1},\ldots,\eta_{M}\in
L_{kq}(\Omega;E)$ be independent $E$-valued random variables with vanishing
$k$th moment, $\mathbb{M}^{k}_{\varepsilon}[\eta_{j}]=0$ for all $1\leq j\leq
M$. Then,

$\biggl{\|}\sum_{j=1}^{M}\otimes^{k}\eta_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}\leq
4kK_{q,p}\tau_{p}(E)\,\Biggl{(}\sum_{j=1}^{M}\|\eta_{j}\|_{L_{kq}(\Omega;E)}^{kp}\Biggr{)}^{\nicefrac{{1}}{{p}}}\\!.$
(3.18)

######  Remark 3.18.

Optimality of the convergence rate $1-\frac{1}{p}$ in (3.13) is ultimately
related to the question whether it is _necessary_ that the Banach space $E$
has Rademacher type $p\in[1,2]$ for (3.18) to hold for all finite sequences
${\eta_{1},\ldots,\eta_{M}\in L_{kq}(\Omega;E)}$ of independent $E$-valued
random variables with vanishing $k$th moment.

For the first moment, $k=1$, it is evident that the choice
$\eta_{j}:=r_{j}x_{j}$ in (3.18), where $(r_{j})_{j\in\mathbb{N}}$ is a
Rademacher family on $(\Omega,\mathcal{A},\mathbb{P})$ and
$x_{1},x_{2},\ldots\in E$, implies that the Banach space $E$ has Rademacher
type $p$. However, for higher-order moments, this question is more complex due
to the injective tensor norm on the left-hand side. For odd orders
$k\in\mathbb{N}$ and the space $E:=\ell_{1}$ of summable real-valued sequences
(which only has Rademacher type $p=1$), the choice $\eta_{j}:=r_{j}e_{j}$
shows that (3.18) cannot hold for any $p>1$. Here, $(e_{j})_{j\in\mathbb{N}}$
denote the standard unit vectors in $\ell_{1}$. In addition, the classical
Khintchine inequalities (3.12) imply that, for any Banach space $E$, the
convergence rate in (3.13) cannot be better that $\nicefrac{{1}}{{2}}$ (one
may take, e.g., $\xi_{j}:=g_{j}\,x$, where $(g_{j})_{j\in\mathbb{N}}$ is an
orthogaussian family on $(\Omega,\mathcal{A},\mathbb{P})$ and $x\neq 0$ is a
non-zero vector in $E$). Sharpness of the rate $1-\frac{1}{p}$ in (3.13) and
necessity of the Rademacher type $p$ for (3.18) for the case $p\in(1,2)$
remains an open question.

The next lemma complements Theorem 3.16 when deriving convergence rates of
single-level Monte Carlo methods for approximating injective $k$th moments.

######  Lemma 3.19.

Let $k\in\mathbb{N}$ and suppose that $\eta,\xi\in L_{k}(\Omega;E)$. Then,

$\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta]-\mathbb{M}^{k}_{\varepsilon}[\xi]\bigr{\|}_{\varepsilon_{s}}\leq\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta]-\mathbb{M}^{k}_{\varepsilon}[\xi]\bigr{\|}_{\varepsilon}\leq\|\eta-\xi\|_{L_{k}(\Omega;E)}\sum_{i=0}^{k-1}\Bigl{[}\|\eta\|_{L_{k}(\Omega;E)}^{i}\|\xi\|_{L_{k}(\Omega;E)}^{k-i-1}\Bigr{]}.$

###### Proof.

The first inequality of the assertion is trivial. We next note that also the
remaining relation is evident in the case $k=1$, since

$\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta]-\mathbb{M}^{k}_{\varepsilon}[\xi]\bigr{\|}_{\varepsilon}=\bigl{\|}\mathbb{E}[\eta-\xi]\bigr{\|}_{E}\leq\mathbb{E}\bigl{[}\|\eta-\xi\|_{E}\bigr{]}=\|\eta-\xi\|_{L_{k}(\Omega;E)}\quad\text{if}\quad
k=1.$

We now assume that $k\geq 2$ and observe that

$\displaystyle{\otimes^{k}}\eta-{\otimes^{k}}\xi$
$\displaystyle=\sum_{i=0}^{k-1}\Bigl{[}\bigl{(}\otimes^{i+1}\eta\bigr{)}\otimes\bigl{(}\otimes^{k-(i+1)}\xi\bigr{)}-\bigl{(}\otimes^{i}\eta\bigr{)}\otimes\bigl{(}\otimes^{k-i}\xi\bigr{)}\Bigr{]}$
$\displaystyle=\sum_{i=0}^{k-1}\Bigl{[}\bigl{(}\otimes^{i}\eta\bigr{)}\otimes(\eta-\xi)\otimes\bigl{(}\otimes^{k-(i+1)}\xi\bigr{)}\Bigr{]}.$

Therefore, we may estimate as follows,

$\displaystyle\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta]-\mathbb{M}^{k}_{\varepsilon}[\xi]\bigr{\|}_{\varepsilon}=\bigl{\|}\mathbb{E}\bigl{[}{\otimes^{k}}\eta-{\otimes^{k}}\xi\bigr{]}\bigr{\|}_{\varepsilon}\leq\mathbb{E}\Bigl{[}\bigl{\|}{\otimes^{k}}\eta-{\otimes^{k}}\xi\bigr{\|}_{\varepsilon}\Bigr{]}$
$\displaystyle\leq\mathbb{E}\Biggl{[}\,\sum_{i=0}^{k-1}\bigl{\|}\bigl{(}\otimes^{i}\eta\bigr{)}\otimes(\eta-\xi)\otimes\bigl{(}\otimes^{k-i-1}\xi\bigr{)}\bigr{\|}_{\varepsilon}\Biggr{]}=\sum_{i=0}^{k-1}\mathbb{E}\Bigl{[}\|\eta\|_{E}^{i}\|\eta-\xi\|_{E}\|\xi\|_{E}^{k-i-1}\Bigr{]}.$

Combined with the Hölder inequality this completes the proof, since

$\displaystyle\mathbb{E}\Bigl{[}\|\eta-\xi\|_{E}\|\xi\|_{E}^{k-1}\Bigr{]}$
$\displaystyle\leq\bigl{(}\mathbb{E}\bigl{[}\|\eta-\xi\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{1}{k}}\bigl{(}\mathbb{E}\bigl{[}\|\xi\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{k-1}{k}},$
$\displaystyle\mathbb{E}\Bigl{[}\|\eta\|_{E}^{k-1}\|\eta-\xi\|_{E}\Bigr{]}$
$\displaystyle\leq\bigl{(}\mathbb{E}\bigl{[}\|\eta-\xi\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{1}{k}}\bigl{(}\mathbb{E}\bigl{[}\|\eta\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{k-1}{k}},$

and, whenever $k\geq 3$, we obtain for all $i\in\\{1,\ldots,k-2\\}$

$\displaystyle\mathbb{E}\Bigl{[}\|\eta\|_{E}^{i}\|\eta-\xi\|_{E}\|\xi\|_{E}^{k-i-1}\Bigr{]}\leq\bigl{(}\mathbb{E}\bigl{[}\|\eta\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{i}{k}}\bigl{(}\mathbb{E}\bigl{[}\|\eta-\xi\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{1}{k}}\bigl{(}\mathbb{E}\bigl{[}\|\xi\|_{E}^{k}\bigr{]}\bigr{)}^{\frac{k-i-1}{k}}$

by a triple Hölder inequality with
$\bigl{(}\tfrac{k}{i}\bigr{)}^{-1}+k^{-1}+\bigl{(}\tfrac{k}{k-i-1}\bigr{)}^{-1}=1$.
∎

We are now ready to state the main result of this subsection: an abstract
convergence rate bound in $L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)$ for
single-level Monte Carlo estimation of the injective $k$th moment
$\mathbb{M}^{k}_{\varepsilon}[\eta]$, assuming at our disposal $M$ independent
samples of an approximation $\xi_{1}\in L_{kq}(\Omega;E)$ to $\eta\in
L_{k}(\Omega;E)$.

######  Corollary 3.20.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in[1,2]$. Let
$q\in[p,\infty)$, $k,M\in\mathbb{N}$ and $\xi_{1},\ldots,\xi_{M}\in
L_{kq}(\Omega;E)$ be independent and identically distributed $E$-valued random
variables. Then, for every $U\in\otimes^{k,s}_{\varepsilon_{s}}E$, we have

$\biggl{\|}U-\frac{1}{M}\sum_{j=1}^{M}\otimes^{k}\xi_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}\leq\bigl{\|}U-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}+C^{\,\sf
SL}_{q,p,k}\,M^{-\left(1-\frac{1}{p}\right)}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k},$

where the constant $C^{\,\sf SL}_{q,p,k}$ is defined as in (3.14).

In particular, for all $\eta\in L_{k}(\Omega;E)$, we have

$\displaystyle\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{k}\xi_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle\leq\|\eta-\xi_{1}\|_{L_{k}(\Omega;E)}\sum_{i=0}^{k-1}\Bigl{[}\|\eta\|_{L_{k}(\Omega;E)}^{i}\|\xi_{1}\|_{L_{k}(\Omega;E)}^{k-i-1}\Bigr{]}$
$\displaystyle\quad+C^{\,\sf
SL}_{q,p,k}\,M^{-\left(1-\frac{1}{p}\right)}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k}.$

###### Proof.

By the triangle inequality on $L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)$
we obtain, for every $U\in\otimes^{k,s}_{\varepsilon_{s}}E$,

$\displaystyle\biggl{\|}U-\frac{1}{M}\sum_{j=1}^{M}\otimes^{k}\xi_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle\leq\bigl{\|}U-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}$
$\displaystyle\quad+\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{k}\xi_{j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)},$

and the first claim follows by applying the estimate (3.13) of Theorem 3.16.
Subsequently, we derive the second assertion by combining this result with
Lemma 3.19 which we use to bound the difference of the $k$th moments
$\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta]-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}$.
∎

We close this subsection with a counterexample which shows that the
convergence results for the standard and single-level Monte Carlo estimators
in Theorem 3.16 and Corollary 3.20 can, in general, not hold when measuring
the error in the (symmetric or full) projective tensor norm. More
specifically, we discuss this for second-order moments of Hilbert space valued
random variables, i.e., the random variables take values in a Banach space of
type $p=2$.

######  Example 3.21.

Let $(H,(\,\cdot\,,\,\cdot\,)_{H})$ be a real separable Hilbert space, and let
$(e_{j})_{j\in\mathbb{N}}$ be an orthonormal basis for $H$. For
$n\in\mathbb{N}$, consider the $H$-valued random variable
$\xi_{n}\colon\Omega\to H$ on $(\Omega,\mathcal{A},\mathbb{P})$, whose
(discrete uniform) distribution is defined by

$\forall
i\in\\{1,\ldots,n\\}:\quad\mathbb{P}(\\{\omega\in\Omega:\xi_{n}(\omega)=e_{i}\\})=n^{-1}\\!.$

Then, for all $n\in\mathbb{N}$, both the projective and injective second
moments of $\xi_{n}$ exist,

$\mathbb{M}^{2}_{\pi}[\xi_{n}]=\mathbb{M}^{2}_{\varepsilon}[\xi_{n}]=\mathbb{E}\bigl{[}\xi_{n}\otimes\xi_{n}\bigr{]}=\frac{1}{n}\sum_{i=1}^{n}e_{i}\otimes
e_{i},\quad\bigl{\|}\mathbb{M}^{2}_{\pi}[\xi_{n}]\bigr{\|}_{\pi}=\bigl{\|}\mathbb{M}^{2}_{\pi}[\xi_{n}]\bigr{\|}_{\pi_{s}}=1,$

see (A.2) in Lemma A.1 of Appendix A for the norm identities.

In addition, for all $q\in[1,\infty)$ and every $n\in\mathbb{N}$, we have that

$\|\xi_{n}\|_{L_{q}(\Omega;H)}^{q}=\mathbb{E}\bigl{[}\|\xi_{n}\|_{H}^{q}\bigr{]}=\frac{1}{n}\sum_{i=1}^{n}\|e_{i}\|_{H}^{q}=1.$

We let $q\in[1,\infty)$, $\xi_{n,1},\ldots,\xi_{n,M}$ be $M\in\mathbb{N}$
independent copies of $\xi_{n}$ and estimate

$\displaystyle\mathrm{err}_{q,\pi}^{(n)}:=\biggl{\|}\mathbb{M}^{2}_{\pi}[\xi_{n}]$
$\displaystyle-\frac{1}{M}\sum_{j=1}^{M}\otimes^{2}\xi_{n,j}\biggr{\|}_{L_{q}(\Omega;\otimes_{\pi}^{2}H)}^{q}=\mathbb{E}\Biggl{[}\biggl{\|}\mathbb{M}^{2}_{\pi}[\xi_{n}]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{2}\xi_{n,j}\biggr{\|}_{\pi}^{q}\Biggr{]}$
$\displaystyle=\sum_{\nu_{1}=1}^{n}\cdots\sum_{\nu_{M}=1}^{n}\frac{1}{n^{M}}\biggl{\|}\,\frac{1}{n}\sum_{i=1}^{n}(e_{i}\otimes
e_{i})-\frac{1}{M}\sum_{j=1}^{M}\bigl{(}e_{\nu_{j}}\otimes
e_{\nu_{j}}\bigr{)}\biggr{\|}_{\pi}^{q}$
$\displaystyle\geq\sum_{\begin{subarray}{c}1\leq\nu_{1},\ldots,\nu_{M}\leq
n\\\ \text{pairwise
distinct}\end{subarray}}\frac{1}{n^{M}}\biggl{\|}\,\frac{1}{n}\sum_{i=1}^{n}(e_{i}\otimes
e_{i})-\frac{1}{M}\sum_{j=1}^{M}\bigl{(}e_{\nu_{j}}\otimes
e_{\nu_{j}}\bigr{)}\biggr{\|}_{\pi}^{q}.$

Thus, assuming that $n\geq M$, again by (A.2) in Lemma A.1 we obtain that

$\displaystyle\mathrm{err}_{q,\pi}^{(n)}$
$\displaystyle\geq\sum_{\begin{subarray}{c}1\leq\nu_{1},\ldots,\nu_{M}\leq
n\\\ \text{pairwise
distinct}\end{subarray}}n^{-M}\bigl{[}M\bigl{(}\tfrac{1}{M}-\tfrac{1}{n}\bigr{)}+(n-M)\tfrac{1}{n}\bigr{]}^{q}$
$\displaystyle=2^{q}\bigl{(}1-\tfrac{M}{n}\bigr{)}^{q}n^{-M}[n\cdots(n-M+1)]\geq
2^{q}\bigl{(}1-\tfrac{M}{n}\bigr{)}^{q+M}\\!.$

Given $q\in[1,\infty)$ and $M\in\mathbb{N}$, we choose an integer
$n_{\star}=n_{\star}(q,M)\in\mathbb{N}$ such that

$n_{\star}\geq
M\bigl{(}1-2^{-\nicefrac{{q}}{{(q+M)}}}\bigr{)}^{-1}\qquad\Longrightarrow\qquad\bigl{(}1-\tfrac{M}{n_{\star}}\bigr{)}^{q+M}\geq
2^{-q}.$

This proves that, for all $q\in[1,\infty)$ and every $M\in\mathbb{N}$, there
exists ${n_{\star}=n_{\star}(q,M)\in\mathbb{N}}$ such that

$\biggl{\|}\mathbb{M}^{2}_{\pi}[\xi_{n_{\star}}]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{2}\xi_{n_{\star},j}\biggr{\|}_{L_{q}(\Omega;\otimes_{\pi_{s}}^{2,s}H)}\geq\biggl{\|}\mathbb{M}^{2}_{\pi}[\xi_{n_{\star}}]-\frac{1}{M}\sum_{j=1}^{M}\otimes^{2}\xi_{n_{\star},j}\biggr{\|}_{L_{q}(\Omega;\otimes_{\pi}^{2}H)}\geq
1.$

Since $\|\xi_{n_{\star}}\|_{L_{q}(\Omega;H)}=1$ is also true for all
$q\in[1,\infty)$ and since $H$ has type $p=2$, this shows that an analogue of
(3.13) cannot hold with respect to the (full or symmetric) projective tensor
norm.

###  3.3. Multilevel Monte Carlo estimation

Assuming that $(X_{\ell})_{\ell=1}^{L}$ is a family of $E$-valued random
variables corresponding to $L\in\mathbb{N}$ different refinement _levels_ of
underlying discretization parameters, translating the idea of multilevel Monte
Carlo (MLMC) estimation, as formulated e.g. in [12, p. 5] for means of Hilbert
space valued random variables, to higher-order moments of Banach space valued
random variables results in exploiting the telescopic sum (here: $X_{0}:=0\in
E$)

$\mathbb{E}\bigl{[}{\otimes^{k}}X_{L}\bigr{]}=\mathbb{M}^{k}_{\varepsilon}[X_{L}]=\sum_{\ell=1}^{L}\bigl{(}\mathbb{M}^{k}_{\varepsilon}[X_{\ell}]-\mathbb{M}^{k}_{\varepsilon}[X_{\ell-1}]\bigr{)}=\sum_{\ell=1}^{L}\mathbb{E}\bigl{[}{\otimes^{k}}X_{\ell}-{\otimes^{k}}X_{\ell-1}\bigr{]},$

and estimating $\mathbb{E}[{\otimes^{k}}X_{\ell}-{\otimes^{k}}X_{\ell-1}]$ for
each $1\leq\ell\leq L$ via Monte Carlo sampling instead of
$\mathbb{E}[{\otimes^{k}}X_{L}]$. As we will see in Theorem 3.25 and Remark
3.27, this approach considerably reduces the computational cost.

Evidently, the corresponding error analysis requires a Monte Carlo convergence
result for estimating differences of injective $k$th moments, i.e., for
expected values of the form
$\mathbb{E}[{\otimes^{k}}\eta-{\otimes^{k}}\xi]=\mathbb{M}^{k}_{\varepsilon}[\eta]-\mathbb{M}^{k}_{\varepsilon}[\xi]$,
via standard Monte Carlo methods. This auxiliary result is derived in
Proposition 3.23 by means of the next lemma, Lemma 3.22, which acts as the
analogue of the Rademacher type estimate (3.11) for Rademacher sums of
differences ${\otimes^{k}}x_{j}-{\otimes^{k}}y_{j}$, $1\leq j\leq M$.

######  Lemma 3.22.

Let $(r_{j})_{j=1}^{M}$ be a Rademacher family on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$ with
expectation $\widetilde{\mathbb{E}}$. Assume that $(E,\|\,\cdot\,\|_{E})$ has
Rademacher type ${p\in[1,2]}$, and let ${q\in[p,\infty)}$, $k,M\in\mathbb{N}$
and $x_{1},\ldots,x_{M},y_{1},\ldots,y_{M}\in E$. Then,

$\biggl{\|}\sum_{j=1}^{M}r_{j}\bigl{(}\otimes^{k}x_{j}-\otimes^{k}y_{j}\bigr{)}\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\leq
C^{\,\sf
diff}_{q,p,k}\sum_{i=1}^{k}\binom{k}{i}\Biggl{[}\sum_{j=1}^{M}\|x_{j}-y_{j}\|_{E}^{ip}\|y_{j}\|_{E}^{(k-i)p}\Biggr{]}^{\nicefrac{{1}}{{p}}}\\!.$
(3.19)

Here, $\binom{k}{i}:=\frac{k!}{i!(k-i)!}$ is the binomial coefficient and the
constant $C^{\,\sf diff}_{q,p,k}$ is given by

$C^{\,\sf diff}_{q,p,k}:=16k\sqrt{\pi}K_{q,p}\tau_{p}(E)K_{q,2},$ (3.20)

with the Kahane–Khintchine and type $p$ constants from Definitions 3.7 and
3.10.

###### Proof.

For the proof of (3.19), we assume that $(g_{j})_{j=1}^{M}$,
$(\widetilde{g}_{j})_{j=1}^{M}$ are two independent orthogaussian families on
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$. We
first note that by [47, Lemma 4.5 and (4.8)], applied for the convex function
$t\mapsto t^{q}$ and the Banach space $\otimes^{k,s}_{\varepsilon_{s}}E$, and
by the definition (2.3) of the symmetric injective tensor norm we have that

$\begin{split}\biggl{\|}\sum_{j=1}^{M}r_{j}\bigl{(}\otimes^{k}x_{j}&-\otimes^{k}y_{j}\bigr{)}\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\leq\sqrt{\frac{\pi}{2}}\,\biggl{\|}\sum_{j=1}^{M}g_{j}\bigl{(}\otimes^{k}x_{j}-\otimes^{k}y_{j}\bigr{)}\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&=\sqrt{\frac{\pi}{2}}\,\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}\bigl{(}f(x_{j})^{k}-f(y_{j})^{k}\bigr{)}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\\\
&=\sqrt{\frac{\pi}{2}}\,\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}\sum_{i=1}^{k}\binom{k}{i}f(x_{j}-y_{j})^{i}f(y_{j})^{k-i}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\\\
&\leq\sum_{i=1}^{k}\sqrt{\frac{\pi}{2}}\binom{k}{i}\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}f(x_{j}-y_{j})^{i}f(y_{j})^{k-i}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\\!.\end{split}$
(3.21)

Here, we also used the binomial expansion for
$f(x_{j})^{k}=[f(x_{j}-y_{j})+f(y_{j})]^{k}$ and the triangle inequality on
$L_{q}(\widetilde{\Omega};\mathbb{R})$. We now claim that, for every
$i\in\\{1,\ldots,k-1\\}$,

$\begin{split}&\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}f(x_{j}-y_{j})^{i}f(y_{j})^{k-i}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\\\
&\leq 2\sqrt{2}\,\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}\bigl{(}g_{j}f(x_{j}-y_{j})^{i}\|y_{j}\|_{E}^{k-i}+\widetilde{g}_{j}\|x_{j}-y_{j}\|_{E}^{i}f(y_{j})^{k-i}\bigr{)}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\\!.\end{split}$
(3.22)

To establish (3.22), we set $\delta_{j}:=x_{j}-y_{j}\in E$ for all $1\leq
j\leq M$, and consider for a fixed $i\in\\{1,\dots,k-1\\}$ the following two
real-valued centered Gaussian processes
$\mathcal{G}_{i,1},\mathcal{G}_{i,2}\colon
B_{{E}^{\prime}}\\!\times\widetilde{\Omega}\to\mathbb{R}$, which are indexed
by $f\in B_{{E}^{\prime}}$,

$\displaystyle\mathcal{G}_{i,1}(f)$
$\displaystyle:=\sum_{j=1}^{M}g_{j}f(\delta_{j})^{i}f(y_{j})^{k-i},$ (3.23)
$\displaystyle\mathcal{G}_{i,2}(f)$
$\displaystyle:=\sqrt{2}\,\sum_{j=1}^{M}\bigl{(}g_{j}f(\delta_{j})^{i}\|y_{j}\|_{E}^{k-i}+\widetilde{g}_{j}\|\delta_{j}\|_{E}^{i}f(y_{j})^{k-i}\bigr{)}.$
(3.24)

For all $i\in\\{1,\dots,k-1\\}$ and every $f,h\in B_{{E}^{\prime}}$, we then
obtain by independence of the standard Gaussian random variables
$g_{1},\ldots,g_{M},\widetilde{g}_{1},\ldots,\widetilde{g}_{M}$ the following
estimate,

$\displaystyle\widetilde{\mathbb{E}}\bigl{[}|\mathcal{G}_{i,1}(f)$
$\displaystyle-\mathcal{G}_{i,1}(h)|^{2}\bigr{]}=\sum_{j=1}^{M}\bigl{(}f(\delta_{j})^{i}f(y_{j})^{k-i}-h(\delta_{j})^{i}h(y_{j})^{k-i}\bigr{)}^{2}$
$\displaystyle=\sum_{j=1}^{M}\bigl{(}\bigl{[}f(\delta_{j})^{i}-h(\delta_{j})^{i}\bigr{]}f(y_{j})^{k-i}+h(\delta_{j})^{i}\bigl{[}f(y_{j})^{k-i}-h(y_{j})^{k-i}\bigr{]}\bigr{)}^{2}$
$\displaystyle\leq
2\sum_{j=1}^{M}\Bigl{(}\bigl{[}f(\delta_{j})^{i}-h(\delta_{j})^{i}\bigr{]}^{2}f(y_{j})^{2(k-i)}+h(\delta_{j})^{2i}\bigl{[}f(y_{j})^{k-i}-h(y_{j})^{k-i}\bigr{]}^{2}\Bigr{)}$
$\displaystyle\leq
2\sum_{j=1}^{M}\Bigl{(}\bigl{[}f(\delta_{j})^{i}-h(\delta_{j})^{i}\bigr{]}^{2}\|y_{j}\|_{E}^{2(k-i)}+\|\delta_{j}\|_{E}^{2i}\bigl{[}f(y_{j})^{k-i}-h(y_{j})^{k-i}\bigr{]}^{2}\Bigr{)}$
$\displaystyle=\widetilde{\mathbb{E}}\bigl{[}|\mathcal{G}_{i,2}(f)-\mathcal{G}_{i,2}(h)|^{2}\bigr{]}.$

Furthermore, for every $i\in\\{1,\dots,k-1\\}$, we have

$\mathcal{G}_{i,1}(f)=\sum_{j=1}^{M}g_{j}\Psi_{i}\bigl{(}f(\delta_{j}),f(y_{j})\bigr{)},\quad
f\in B_{{E}^{\prime}\\!}.$

Here, $(t_{1},t_{2})\mapsto\Psi_{i}(t_{1},t_{2}):=t_{1}^{i}\,t_{2}^{k-i}$ is
continuous on $\mathbb{R}^{2}$ and satisfies $\Psi_{i}(0,0)=0$ for all
$i\in\\{1,\dots,k-1\\}$.

We thus may apply the comparison result derived in Lemma B.2 (see Appendix B)
for every ${i\in\\{1,\ldots,k-1\\}}$, which shows that, for all
$q\in[1,\infty)$,

$\displaystyle\widetilde{\mathbb{E}}\Bigl{[}\bigl{(}\sup\nolimits_{f\in
B_{{E}^{\prime}}\\!}|\mathcal{G}_{i,1}(f)|\bigr{)}^{q}\Bigr{]}\leq
2^{q}\,\widetilde{\mathbb{E}}\Bigl{[}\bigl{(}\sup\nolimits_{f\in
B_{{E}^{\prime}}\\!}|\mathcal{G}_{i,2}(f)|\bigr{)}^{q}\Bigr{]}.$

Taking the $q$th root on both sides of this inequality and inserting the
definitions of $\mathcal{G}_{i,1}$ and $\mathcal{G}_{i,2}$ from (3.23)–(3.24)
completes the proof of (3.22).

Next, combining (3.21) with (3.22) and the triangle inequality on
$L_{q}(\widetilde{\Omega};\mathbb{R})$ yields

$\displaystyle\biggl{\|}\sum_{j=1}^{M}r_{j}\bigl{(}\otimes^{k}x_{j}-\otimes^{k}y_{j}\bigr{)}\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\leq\sum_{i=1}^{k-1}\widetilde{C}_{k,i}\,\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}f(\delta_{j})^{i}\|y_{j}\|_{E}^{k-i}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}$
$\displaystyle+\sqrt{\frac{\pi}{2}}\,\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}g_{j}f(\delta_{j})^{k}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}+\sum_{i=1}^{k-1}\widetilde{C}_{k,i}\,\Biggl{\|}\sup_{f\in
B_{{E}^{\prime}}}\biggl{|}\sum_{j=1}^{M}\widetilde{g}_{j}\|\delta_{j}\|_{E}^{i}f(y_{j})^{k-i}\biggr{|}\Biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\\!,$

where we set $\widetilde{C}_{k,i}:=2\sqrt{\pi}\binom{k}{i}$. By noting that
$\sqrt{\pi}/\sqrt{2}\leq\widetilde{C}_{k,k}$ and estimating the
$L_{q}(\widetilde{\Omega};\mathbb{R})$-norms on the right-hand side for all
$i\in\\{1,\ldots,k\\}$ using Proposition 3.4(i), with $k_{j}=i$ and the
vectors $\|y_{j}\|_{E}^{\frac{k-i}{i}}\delta_{j}$ (respectively, for every
$i\in\\{1,\ldots,k-1\\}$ with $k_{j}=k-i$ and
$\|\delta_{j}\|_{E}^{\frac{i}{k-i}}y_{j}$) for all $1\leq j\leq M$, we find
that

$\displaystyle\biggl{\|}\sum_{j=1}^{M}r_{j}\bigl{(}\otimes^{k}x_{j}-\otimes^{k}y_{j}\bigr{)}$
$\displaystyle\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\leq\sum_{i=1}^{k}\widetilde{C}_{k,i}\,4i\,\biggl{\|}\sum_{j=1}^{M}g_{j}\|\delta_{j}\|_{E}^{i-1}\|y_{j}\|_{E}^{k-i}\delta_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};E)}$
$\displaystyle+\sum_{i=1}^{k-1}\widetilde{C}_{k,i}\,4(k-i)\,\biggl{\|}\sum_{j=1}^{M}\widetilde{g}_{j}\|\delta_{j}\|_{E}^{i}\|y_{j}\|_{E}^{k-i-1}y_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};E)}\\!.$

Finally, since $q\in[p,\infty)$ is assumed, we may use Proposition 3.14 for
the independent, centered $E$-valued random variables

$\eta_{j}:=g_{j}\|\delta_{j}\|_{E}^{i-1}\|y_{j}\|_{E}^{k-i}\delta_{j}\quad\text{resp.}\quad\widetilde{\eta}_{j}:=\widetilde{g}_{j}\|\delta_{j}\|_{E}^{i}\|y_{j}\|_{E}^{k-i-1}y_{j},\qquad
1\leq j\leq M,$

to conclude that (recall the definitions
$\widetilde{C}_{k,i}=2\sqrt{\pi}\binom{k}{i}$ and $\delta_{j}=x_{j}-y_{j}$)

$\displaystyle\Biggl{\|}\sum_{j=1}^{M}r_{j}$
$\displaystyle\bigl{(}\otimes^{k}x_{j}-\otimes^{k}y_{j}\bigr{)}\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle\leq\sum_{i=1}^{k}\left[8k\sqrt{\pi}\binom{k}{i}\,2K_{q,p}\tau_{p}(E)\,\|g_{1}\|_{L_{q}(\widetilde{\Omega};\mathbb{R})}\Biggl{[}\sum_{j=1}^{M}\|x_{j}-y_{j}\|_{E}^{ip}\|y_{j}\|_{E}^{(k-i)p}\Biggr{]}^{\nicefrac{{1}}{{p}}}\right],$

which completes the proof of (3.19), since
$\|g_{1}\|_{L_{q}(\widetilde{\Omega};\mathbb{R})}\leq
K_{q,2}\|g_{1}\|_{L_{2}(\widetilde{\Omega};\mathbb{R})}=K_{q,2}$ follows from
(3.10), and $C^{\,\sf diff}_{q,p,k}=16k\sqrt{\pi}K_{q,p}\tau_{p}(E)K_{q,2}$ by
(3.20). ∎

######  Proposition 3.23.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in[1,2]$. Let
$q\in[p,\infty)$, $k,M\in\mathbb{N}$ and
$\eta_{1},\ldots,\eta_{M},\xi_{1},\ldots,\xi_{M}\in L_{kq}(\Omega;E)$ be
$E$-valued random variables such that the tuples
$(\eta_{1},\xi_{1}),\ldots,(\eta_{M},\xi_{M})$ are independent and identically
distributed. Then,

$\displaystyle\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]$
$\displaystyle-\frac{1}{M}\sum_{j=1}^{M}\bigl{(}\otimes^{k}\eta_{j}-\otimes^{k}\xi_{j}\bigr{)}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle\leq 2C^{\,\sf
diff}_{q,p,k}\,M^{-\left(1-\frac{1}{p}\right)}\sum_{i=1}^{k}\Bigl{[}{\textstyle\binom{k}{i}}\|\eta_{1}-\xi_{1}\|_{L_{kq}(\Omega;E)}^{i}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k-i}\Bigr{]}$
$\displaystyle\quad+2B_{q}\,M^{-\nicefrac{{1}}{{2}}}\|\eta_{1}-\xi_{1}\|_{L_{k}(\Omega;E)}\sum_{i=0}^{k-1}\Bigl{[}\|\eta_{1}\|_{L_{k}(\Omega;E)}^{i}\|\xi_{1}\|_{L_{k}(\Omega;E)}^{k-i-1}\Bigr{]},$

where $B_{q},C^{\,\sf diff}_{q,p,k}\in(0,\infty)$ are defined as in (3.12) and
(3.20), respectively.

###### Proof.

We proceed similarly as in the proof of Theorem 3.16. We pick a Rademacher
family $(r_{j})_{j=1}^{M}$ on a complete probability space
$(\widetilde{\Omega},\widetilde{\mathcal{A}},\widetilde{\mathbb{P}})$, and
define the following random variables on the product probability space
$(\Omega\times\widetilde{\Omega},\mathcal{A}\otimes\widetilde{\mathcal{A}},\mathbb{P}\otimes\widetilde{\mathbb{P}})$:
For every $j\in\\{1,\ldots,M\\}$, we set

$\boldsymbol{\eta}_{j}(\omega,\widetilde{\omega}):=\eta_{j}(\omega),\quad\boldsymbol{\xi}_{j}(\omega,\widetilde{\omega}):=\xi_{j}(\omega),\quad\boldsymbol{r}_{j}(\omega,\widetilde{\omega}):=r_{j}(\widetilde{\omega})\qquad\forall(\omega,\widetilde{\omega})\in\Omega\times\widetilde{\Omega},$

where we note that $(\boldsymbol{r}_{j})_{j=1}^{M}$ is a Rademacher family on
$(\Omega\times\widetilde{\Omega},\mathcal{A}\otimes\widetilde{\mathcal{A}},\mathbb{P}\otimes\widetilde{\mathbb{P}})$,
and that
$(\boldsymbol{\eta}_{1},\boldsymbol{\xi}_{1}),\ldots,(\boldsymbol{\eta}_{M},\boldsymbol{\xi}_{M}),\boldsymbol{r}_{1},\ldots,\boldsymbol{r}_{M}$
are independent. Furthermore, the random variables
$\otimes^{k}\boldsymbol{\eta}_{j}-\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]+\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]$
are centered for all $1\leq j\leq M$ so that by Lemma 3.6 and by the triangle
inequality on
$L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)$ we
find that

$\begin{split}\biggl{\|}\sum_{j=1}^{M}\bigl{(}\otimes^{k}\eta_{j}&-\otimes^{k}\xi_{j}-\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]+\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&=\biggl{\|}\sum_{j=1}^{M}\bigl{(}\otimes^{k}\boldsymbol{\eta}_{j}-\otimes^{k}\boldsymbol{\xi}_{j}-\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]+\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&\leq
2\,\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\bigl{(}\otimes^{k}\boldsymbol{\eta}_{j}-\otimes^{k}\boldsymbol{\xi}_{j}\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}\\\
&\quad+2\,\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\bigl{(}\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}=:2\text{(A)}+2\text{(B)}.\end{split}$
(3.25)

For term (A) we use Fubini’s theorem as well as Lemma 3.22 to find that

(A)
$\displaystyle=\biggl{\|}\sum_{j=1}^{M}\boldsymbol{r}_{j}\bigl{(}\otimes^{k}\boldsymbol{\eta}_{j}-\otimes^{k}\boldsymbol{\xi}_{j}\bigr{)}\biggr{\|}_{L_{q}(\Omega\times\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle=\Biggl{(}\int_{\Omega}\,\biggl{\|}\sum_{j=1}^{M}r_{j}(\,\cdot\,)\bigl{(}\otimes^{k}\eta_{j}(\omega)-\otimes^{k}\xi_{j}(\omega)\bigr{)}\biggr{\|}_{L_{q}(\widetilde{\Omega};\otimes^{k,s}_{\varepsilon_{s}}E)}^{q}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}$
$\displaystyle\leq C^{\,\sf
diff}_{q,p,k}\Biggl{(}\int_{\Omega}\,\Biggl{|}\sum_{i=1}^{k}\binom{k}{i}\biggl{[}\sum_{j=1}^{M}\|\eta_{j}(\omega)-\xi_{j}(\omega)\|_{E}^{ip}\|\xi_{j}(\omega)\|_{E}^{(k-i)p}\biggr{]}^{\nicefrac{{1}}{{p}}}\,\Biggr{|}^{q}\mathrm{d}\mathbb{P}(\omega)\Biggr{)}^{\nicefrac{{1}}{{q}}}$

and, hence,

$\text{(A)}\leq C^{\,\sf
diff}_{q,p,k}\,\Biggl{\|}\sum_{i=1}^{k}\binom{k}{i}\biggl{[}\sum_{j=1}^{M}\|\eta_{j}-\xi_{j}\|_{E}^{ip}\|\xi_{j}\|_{E}^{(k-i)p}\biggr{]}^{\nicefrac{{1}}{{p}}}\Biggr{\|}_{L_{q}(\Omega;\mathbb{R})}.$

Next, we use the triangle inequality on $L_{q}(\Omega;\mathbb{R})$ as well as
the fact that $q\in[p,\infty)$ so that also on
$L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})$ we may apply the triangle
inequality and conclude

(A) $\displaystyle\leq C^{\,\sf
diff}_{q,p,k}\,\sum_{i=1}^{k}\binom{k}{i}\biggl{\|}\sum_{j=1}^{M}\|\eta_{j}-\xi_{j}\|_{E}^{ip}\|\xi_{j}\|_{E}^{(k-i)p}\biggr{\|}_{L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})}^{\nicefrac{{1}}{{p}}}$
$\displaystyle\leq C^{\,\sf
diff}_{q,p,k}\,\sum_{i=1}^{k}\binom{k}{i}\Biggl{(}\sum_{j=1}^{M}\Bigl{\|}\|\eta_{j}-\xi_{j}\|_{E}^{ip}\|\xi_{j}\|_{E}^{(k-i)p}\Bigr{\|}_{L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})}\Biggr{)}^{\nicefrac{{1}}{{p}}}$
$\displaystyle=C^{\,\sf
diff}_{q,p,k}\,M^{\nicefrac{{1}}{{p}}}\sum_{i=1}^{k}\binom{k}{i}\Bigl{\|}\|\eta_{1}-\xi_{1}\|_{E}^{ip}\|\xi_{1}\|_{E}^{(k-i)p}\Bigr{\|}_{L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})}^{\nicefrac{{1}}{{p}}},$

where the last step follows from the identical distribution of
$(\eta_{1},\xi_{1}),\ldots,(\eta_{M},\xi_{M})$. In addition, we observe that,
for every $i\in\\{1,\ldots,k-1\\}$, by Hölder’s inequality

$\displaystyle\Bigl{\|}$
$\displaystyle\|\eta_{1}-\xi_{1}\|_{E}^{ip}\|\xi_{1}\|_{E}^{(k-i)p}\Bigr{\|}_{L_{\nicefrac{{q}}{{p}}}(\Omega;\mathbb{R})}^{\nicefrac{{1}}{{p}}}=\Bigl{(}\mathbb{E}\Bigl{[}\|\eta_{1}-\xi_{1}\|_{E}^{iq}\|\xi_{1}\|_{E}^{(k-i)q}\Bigr{]}\Bigr{)}^{\nicefrac{{1}}{{q}}}$
$\displaystyle\leq\Bigl{(}\bigl{(}\mathbb{E}\bigl{[}\|\eta_{1}-\xi_{1}\|_{E}^{kq}\bigr{]}\bigr{)}^{\frac{i}{k}}\bigl{(}\mathbb{E}\bigl{[}\|\xi_{1}\|_{E}^{kq}\bigr{]}\bigr{)}^{\frac{k-i}{k}}\Bigr{)}^{\nicefrac{{1}}{{q}}}=\|\eta_{1}-\xi_{1}\|_{L_{kq}(\Omega;E)}^{i}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k-i}$

which completes the bound for term (A),

$\text{(A)}\leq C^{\,\sf
diff}_{q,p,k}\,M^{\nicefrac{{1}}{{p}}}\sum_{i=1}^{k}\Bigl{[}{\textstyle\binom{k}{i}}\|\eta_{1}-\xi_{1}\|_{L_{kq}(\Omega;E)}^{i}\|\xi_{1}\|_{L_{kq}(\Omega;E)}^{k-i}\Bigr{]}.$
(3.26)

For term (B) we obtain by the Khintchine inequalities (3.12) and by Lemma 3.19
the following estimate,

$\begin{split}\text{(B)}&=\biggl{\|}\sum_{j=1}^{M}r_{j}\biggr{\|}_{L_{q}(\widetilde{\Omega};\mathbb{R})}\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}\leq
B_{q}\,M^{\nicefrac{{1}}{{2}}}\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[\eta_{1}]-\mathbb{M}^{k}_{\varepsilon}[\xi_{1}]\bigr{\|}_{\varepsilon_{s}}\\\
&\leq
B_{q}\,M^{\nicefrac{{1}}{{2}}}\|\eta_{1}-\xi_{1}\|_{L_{k}(\Omega;E)}\sum_{i=0}^{k-1}\Bigl{[}\|\eta_{1}\|_{L_{k}(\Omega;E)}^{i}\|\xi_{1}\|_{L_{k}(\Omega;E)}^{k-i-1}\Bigr{]}.\end{split}$
(3.27)

The claim now follows by combining (3.25) with the estimates (3.26), (3.27)
for the terms (A) and (B), upon dividing the resulting inequality by $M$. ∎

We are now ready to formulate our convergence result for abstract multilevel
Monte Carlo methods to estimate higher-order statistical moments of Banach
space valued random variables.

######  Theorem 3.24.

Let $(E,\|\,\cdot\,\|_{E})$ be of Rademacher type $p\in[1,2]$,
${q\in[p,\infty)}$ and ${k,L\in\mathbb{N}}$. Suppose further that, for every
$\ell\in\\{1,\ldots,L\\}$, ${X_{\ell}\in L_{kq}(\Omega;E)}$,
$M_{\ell}\in\mathbb{N}$, and $\xi_{\ell,1},\ldots,\xi_{\ell,M_{\ell}}$ are
independent copies of the $\otimes^{k,s}_{\varepsilon_{s}}E$-valued random
variable

$\xi_{\ell}:=\otimes^{k}X_{\ell}-\otimes^{k}X_{\ell-1}\in
L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E),\qquad X_{0}:=0\in E.$

Then, for every $U\in\otimes^{k,s}_{\varepsilon_{s}}E$,

$\displaystyle\biggl{\|}U$
$\displaystyle-\sum_{\ell=1}^{L}\frac{1}{M_{\ell}}\sum_{j=1}^{M_{\ell}}\xi_{\ell,j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}\leq\bigl{\|}U-\mathbb{M}^{k}_{\varepsilon}[X_{L}]\bigr{\|}_{\varepsilon_{s}}$
$\displaystyle+C^{\,\sf
ML}_{q,p,k}\sum_{\ell=1}^{L}\biggl{[}M_{\ell}^{-\left(1-\frac{1}{p}\right)}\|X_{\ell}-X_{\ell-1}\|_{L_{kq}(\Omega;E)}$
$\displaystyle\hskip
48.36967pt\cdot\sum_{i=0}^{k-1}\Bigl{[}\Bigl{(}{\textstyle\binom{k}{i+1}}\|X_{\ell}-X_{\ell-1}\|_{L_{kq}(\Omega;E)}^{i}+\|X_{\ell}\|_{L_{k}(\Omega;E)}^{i}\Bigr{)}\|X_{\ell-1}\|_{L_{kq}(\Omega;E)}^{k-i-1}\Bigr{]}\biggr{]},$

where $C^{\,\sf ML}_{q,p,k}:=2\max\bigl{\\{}C^{\,\sf
diff}_{q,p,k},B_{q}\bigr{\\}}$ and the constants $B_{q},C^{\,\sf
diff}_{q,p,k}\in(0,\infty)$ are defined as in (3.12) and (3.20).

###### Proof.

First note that, for every $\ell\in\\{1,\ldots,L\\}$ the random variables
$\xi_{\ell,1},\ldots,\xi_{\ell,M_{\ell}}$ are identically distributed and we
have that

$\mathbb{E}\Biggl{[}\sum_{\ell=1}^{L}\frac{1}{M_{\ell}}\sum_{j=1}^{M_{\ell}}\xi_{\ell,j}\Biggr{]}=\sum_{\ell=1}^{L}\mathbb{E}[\xi_{\ell}]=\sum_{\ell=1}^{L}\bigl{(}\mathbb{M}^{k}_{\varepsilon}[X_{\ell}]-\mathbb{M}^{k}_{\varepsilon}[X_{\ell-1}]\bigr{)}=\mathbb{M}^{k}_{\varepsilon}[X_{L}].$

Thus, we find by the triangle inequality on
$L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)$ that, for every
$U\in\otimes^{k,s}_{\varepsilon_{s}}E$,

$\displaystyle\biggl{\|}U-\sum_{\ell=1}^{L}\frac{1}{M_{\ell}}\sum_{j=1}^{M_{\ell}}\xi_{\ell,j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}$
$\displaystyle\leq\bigl{\|}U-\mathbb{M}^{k}_{\varepsilon}[X_{L}]\bigr{\|}_{\varepsilon_{s}}+\sum_{\ell=1}^{L}\mathrm{err}^{\sf\,SL}_{q,\varepsilon_{s}}(\xi_{\ell}),$

where, for $\ell\in\\{1,\ldots,L\\}$, we define

$\mathrm{err}^{\sf\,SL}_{q,\varepsilon_{s}}(\xi_{\ell}):=\biggl{\|}\mathbb{E}[\xi_{\ell}]-\frac{1}{M_{\ell}}\sum_{j=1}^{M_{\ell}}\xi_{\ell,j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}.$

For every $\ell\in\\{1,\ldots,L\\}$, we let the tuples
$(X_{\ell-1,1},X_{\ell,1}),\ldots,(X_{\ell-1,M_{\ell}},X_{\ell,M_{\ell}})$ be
$M_{\ell}$ independent copies of $(X_{\ell-1},X_{\ell})$ and observe that

$\displaystyle\mathrm{err}^{\sf\,SL}_{q,\varepsilon_{s}}(\xi_{\ell})=\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[X_{\ell}]-\mathbb{M}^{k}_{\varepsilon}[X_{\ell-1}]-\frac{1}{M_{\ell}}\sum_{j=1}^{M_{\ell}}\bigl{(}\otimes^{k}X_{\ell,j}-\otimes^{k}X_{\ell-1,j}\bigr{)}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}.$

We are thus in the position to apply Proposition 3.23 on every level
$\ell\in\\{1,\ldots,L\\}$,

$\displaystyle\mathrm{err}^{\sf\,SL}_{q,\varepsilon_{s}}(\xi_{\ell})$
$\displaystyle\leq 2C^{\,\sf
diff}_{q,p,k}\,M_{\ell}^{-\left(1-\frac{1}{p}\right)}\sum_{i=1}^{k}\Bigl{[}{\textstyle\binom{k}{i}}\|X_{\ell}-X_{\ell-1}\|_{L_{kq}(\Omega;E)}^{i}\|X_{\ell-1}\|_{L_{kq}(\Omega;E)}^{k-i}\Bigr{]}$
$\displaystyle\quad+2B_{q}\,M_{\ell}^{-\nicefrac{{1}}{{2}}}\|X_{\ell}-X_{\ell-1}\|_{L_{k}(\Omega;E)}\sum_{i=0}^{k-1}\Bigl{[}\|X_{\ell}\|_{L_{k}(\Omega;E)}^{i}\|X_{\ell-1}\|_{L_{k}(\Omega;E)}^{k-i-1}\Bigr{]},$

which after recalling that $p\in[1,2]$ and $q\geq p\geq 1$ as well as
combining the two sums completes the proof of the assertion. ∎

The error estimate of Theorem 3.24 facilitates optimizing the number of levels
$L$ as well as the number of samples on each level, $M_{1},\ldots,M_{L}$, to
reduce the computational cost for achieving a target accuracy $\epsilon>0$ of
the MLMC estimator in the
$L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)$-norm. This optimization is
subject of the following “$\alpha\beta\gamma$ theorem”.

######  Theorem 3.25.

Assume that $(E,\|\,\cdot\,\|_{E})$ is of Rademacher type $p\in(1,2]$. Let
$q\in[p,\infty)$, $k\in\mathbb{N}$, $X\in L_{k}(\Omega;E)$,
$(X_{\ell})_{\ell\in\mathbb{N}}\subset L_{kq}(\Omega;E)$ be a sequence of
$E$-valued random variables and, for every $\ell\in\mathbb{N}$, define

$\xi_{\ell}:=\otimes^{k}X_{\ell}-\otimes^{k}X_{\ell-1}\in
L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E),\qquad X_{0}:=0\in E.$ (3.28)

For $\ell\in\mathbb{N}$, let $\mathcal{C}_{\ell}$ denote the cost (number of
floating point operations) to generate one sample of the random variable
$\xi_{\ell}$ in (3.28), and suppose that there exist a sequence
$(N_{\ell})_{\ell\in\mathbb{N}}$ of positive integers and constants
${\alpha,\beta,\gamma,C_{\alpha},C_{\beta},C_{\gamma},C_{\sf
stab}\in(0,\infty)}$, $A\in(1,\infty)$ such that $N_{\ell}\eqsim A^{\ell}$ for
all $\ell\in\mathbb{N}$ and, moreover,

$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[X]-\mathbb{M}^{k}_{\varepsilon}[X_{\ell}]\bigr{\|}_{\varepsilon_{s}}$
$\displaystyle\leq C_{\alpha}N_{\ell}^{-\alpha}\\!,\qquad$ ($\alpha$)
$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\|X_{\ell}-X_{\ell-1}\|_{L_{kq}(\Omega;E)}$ $\displaystyle\leq
C_{\beta}N_{\ell}^{-\beta}\\!,\qquad$ ($\beta$)
$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\mathcal{C}_{\ell}$ $\displaystyle\leq
C_{\gamma}N_{\ell}^{\gamma},\qquad$ ($\gamma$)
$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\max\bigl{\\{}\|X\|_{L_{k}(\Omega;E)},\,$
$\displaystyle\|X_{\ell}\|_{L_{kq}(\Omega;E)}\bigr{\\}}\leq C_{\sf
stab}.\quad$ ($\sf stab$)

For each $\ell\in\mathbb{N}$, let $(\xi_{\ell,j})_{j\in\mathbb{N}}\subset
L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)$ be a sequence of independent
copies of the $\otimes^{k,s}_{\varepsilon_{s}}E$-valued random variable
$\xi_{\ell}$ in (3.28).

Then, for every $\epsilon\in(0,\nicefrac{{1}}{{2}}]$, there exist integers
$L\in\mathbb{N}$ and $M_{1},\ldots,M_{L}\in\mathbb{N}$ such that the
$L_{q}$-accuracy $\epsilon$ of the multilevel Monte Carlo estimator for
$\mathbb{M}^{k}_{\varepsilon}[X]$,

$\mathrm{err}^{k,\sf
ML}_{q,\varepsilon_{s}}(X):=\biggl{\|}\mathbb{M}^{k}_{\varepsilon}[X]-\sum_{\ell=1}^{L}\frac{1}{M_{\ell}}\sum_{j=1}^{M_{\ell}}\xi_{\ell,j}\biggr{\|}_{L_{q}(\Omega;\otimes^{k,s}_{\varepsilon_{s}}E)}<\epsilon,$
($\epsilon$)

can be achieved at computational costs of the order

$\mathcal{C}^{k,\sf
ML}_{q,\varepsilon_{s}}(X)\lesssim_{(\alpha,\beta,\gamma,A,p,q)}\begin{cases}\epsilon^{-\frac{\gamma}{\alpha}}+\epsilon^{-p^{\prime}}&\text{if}\quad\beta
p^{\prime}>\gamma,\\\
\epsilon^{-\frac{\gamma}{\alpha}}+\epsilon^{-p^{\prime}}|\log_{A}\epsilon|^{p^{\prime}+1}&\text{if}\quad\beta
p^{\prime}=\gamma,\\\
\epsilon^{-\frac{\gamma}{\alpha}}+\epsilon^{-p^{\prime}-\frac{\gamma-\beta
p^{\prime}}{\alpha}}&\text{if}\quad\beta p^{\prime}<\gamma,\end{cases}$
($\mathcal{C}$)

where $p^{\prime}\in[2,\infty)$ is such that
$\tfrac{1}{p}+\tfrac{1}{p^{\prime}}=1$. The constant implied in $\lesssim$ may
also depend on the constants $C_{\alpha},C_{\beta},C_{\gamma}$ and
$C_{\mathsf{stab}}$ from the assumptions above.

###### Proof.

We will show by explicit construction that, for every
$\epsilon\in(0,\nicefrac{{1}}{{2}}]$, assumptions ($\alpha$), ($\beta$),
($\gamma$) and ($\sf stab$) allow to choose the algorithmic steering
parameters $L\in\mathbb{N}$ and $M_{1},\ldots,M_{L}\in\mathbb{N}$ so that
($\epsilon$) holds with cost ($\mathcal{C}$).

By Theorem 3.24 and by assumptions ($\alpha$), ($\beta$), ($\sf stab$) we
obtain the estimate

$\displaystyle\mathrm{err}^{k,\sf
ML}_{q,\varepsilon_{s}}(X)\leq\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[X]-\mathbb{M}^{k}_{\varepsilon}[X_{L}]\bigr{\|}_{\varepsilon_{s}}+C^{\,\sf
ML}_{q,p,k}\sum_{\ell=1}^{L}\biggl{[}M_{\ell}^{-\left(1-\frac{1}{p}\right)}\|X_{\ell}-X_{\ell-1}\|_{L_{kq}(\Omega;E)}$
$\displaystyle\quad\hskip
51.21495pt\cdot\sum_{i=0}^{k-1}\Bigl{[}\Bigl{(}{\textstyle\binom{k}{i+1}}\|X_{\ell}-X_{\ell-1}\|_{L_{kq}(\Omega;E)}^{i}+\|X_{\ell}\|_{L_{k}(\Omega;E)}^{i}\Bigr{)}\|X_{\ell-1}\|_{L_{kq}(\Omega;E)}^{k-i-1}\Bigr{]}\biggr{]}$
$\displaystyle\leq C_{\alpha}N_{L}^{-\alpha}+C^{\,\sf
ML}_{q,p,k}C_{\beta}\sum_{\ell=1}^{L}\biggl{[}M_{\ell}^{-\left(1-\frac{1}{p}\right)}N_{\ell}^{-\beta}\sum_{i=0}^{k-1}\Bigl{[}\Bigl{(}{\textstyle\binom{k}{i+1}}C_{\beta}^{i}+C_{\sf
stab}^{i}\Bigr{)}C_{\sf stab}^{k-i-1}\Bigr{]}\biggr{]}$ $\displaystyle\leq
C_{\alpha}N_{L}^{-\alpha}+C_{\star}\sum_{\ell=1}^{L}\Bigl{[}M_{\ell}^{-\nicefrac{{1}}{{p^{\prime}}}}N_{\ell}^{-\beta}\Bigr{]},$

where $C^{\,\sf ML}_{q,p,k}\in(0,\infty)$ is as in Theorem 3.24, and
$C_{\star}=C_{\star}(k,p,q,C_{\beta},C_{\sf stab})\in(0,\infty)$ is defined by

$C_{\star}:=C^{\,\sf ML}_{q,p,k}C_{\beta}\bigl{(}k\,C_{\sf
stab}^{k-1}+2^{k}\max\bigl{\\{}C_{\beta}^{k-1},C_{\sf
stab}^{k-1}\bigr{\\}}\bigr{)},$

since

$\displaystyle\sum_{i=0}^{k-1}\Bigl{[}\Bigl{(}{\textstyle\binom{k}{i+1}}C_{\beta}^{i}+C_{\sf
stab}^{i}\Bigr{)}C_{\sf stab}^{k-i-1}\Bigr{]}$ $\displaystyle\leq k\,C_{\sf
stab}^{k-1}+\max\bigl{\\{}C_{\beta}^{k-1},C_{\sf
stab}^{k-1}\bigr{\\}}\sum_{i=0}^{k-1}{\textstyle\binom{k}{i+1}}$
$\displaystyle\leq k\,C_{\sf
stab}^{k-1}+2^{k}\max\bigl{\\{}C_{\beta}^{k-1},C_{\sf stab}^{k-1}\bigr{\\}}.$

Choose $L\in\mathbb{N}$ as the smallest integer such that
$N_{L}^{-\alpha}<\min\\{C_{\alpha}^{-1},1\\}\,\tfrac{\epsilon}{2}$ holds and,
for every $\ell\in\\{1,\ldots,L\\}$, let $M_{\ell}\in\mathbb{N}$ be defined as
the smallest integer satisfying

$M_{\ell}\geq C_{\star}^{p^{\prime}}N_{L}^{\alpha
p^{\prime}}S_{L}^{p^{\prime}}N_{\ell}^{-\frac{(\beta+\gamma)p^{\prime}}{p^{\prime}+1}},\qquad\text{where}\qquad
S_{L}:=\sum_{\ell=1}^{L}N_{\ell}^{\frac{\gamma-\beta
p^{\prime}}{p^{\prime}+1}}.$

Note that the magnitude of $S_{L}$ behaves asymptotically (for $L$ large) as

$S_{L}=\sum_{\ell=1}^{L}N_{\ell}^{\frac{\gamma-\beta
p^{\prime}}{p^{\prime}+1}}\eqsim_{(\beta,\gamma,A,p)}\begin{cases}1&\text{if}\quad\beta
p^{\prime}>\gamma,\\\ L&\text{if}\quad\beta p^{\prime}=\gamma,\\\
N_{L}^{\frac{\gamma-\beta p^{\prime}}{p^{\prime}+1}}&\text{if}\quad\beta
p^{\prime}<\gamma.\end{cases}$ (3.29)

For this choice of $L$ and $M_{1},\ldots,M_{L}$, we can bound the error as
follows,

$\displaystyle\mathrm{err}^{k,\sf ML}_{q,\varepsilon_{s}}(X)$
$\displaystyle<C_{\alpha}C_{\alpha}^{-1}\frac{\epsilon}{2}+C_{\star}\sum_{\ell=1}^{L}\biggl{[}C_{\star}^{-1}N_{L}^{-\alpha}S_{L}^{-1}N_{\ell}^{\frac{\beta+\gamma}{p^{\prime}+1}}N_{\ell}^{-\beta}\biggr{]}$
$\displaystyle=\frac{\epsilon}{2}+N_{L}^{-\alpha}S_{L}^{-1}\sum_{\ell=1}^{L}N_{\ell}^{\frac{\gamma-\beta
p^{\prime}}{p^{\prime}+1}}=\frac{\epsilon}{2}+N_{L}^{-\alpha}<\epsilon.$

For the total cost, we first compute

$\displaystyle\mathcal{C}^{k,\sf ML}_{q,\varepsilon_{s}}$
$\displaystyle(X)\eqsim\sum_{\ell=1}^{L}\mathcal{C}_{\ell}M_{\ell}\leq
C_{\gamma}\sum_{\ell=1}^{L}N_{\ell}^{\gamma}\biggl{(}1+C_{\star}^{p^{\prime}}N_{L}^{\alpha
p^{\prime}}S_{L}^{p^{\prime}}N_{\ell}^{-\frac{(\beta+\gamma)p^{\prime}}{p^{\prime}+1}}\biggr{)}$
$\displaystyle\leq
C_{\gamma}\sum_{\ell=1}^{L}N_{\ell}^{\gamma}+C_{\gamma}C_{\star}^{p^{\prime}}N_{L}^{\alpha
p^{\prime}}S_{L}^{p^{\prime}}\sum_{\ell=1}^{L}N_{\ell}^{\frac{\gamma-\beta
p^{\prime}}{p^{\prime}+1}}=C_{\gamma}\sum_{\ell=1}^{L}N_{\ell}^{\gamma}+C_{\gamma}C_{\star}^{p^{\prime}}N_{L}^{\alpha
p^{\prime}}S_{L}^{p^{\prime}+1}\\!.$

By the choice of $L$ we have $A^{L}\eqsim
N_{L}\eqsim\epsilon^{-\nicefrac{{1}}{{\alpha}}}$ and, since
$\epsilon\in(0,\nicefrac{{1}}{{2}}]$, we find that
$L\eqsim_{\alpha}|\log_{A}\epsilon|$. Thus, using (3.29) we conclude that the
computational cost,

$\displaystyle\mathcal{C}^{k,\sf
ML}_{q,\varepsilon_{s}}(X)\lesssim_{(\beta,\gamma,A,p,q)}\begin{cases}N_{L}^{\gamma}+N_{L}^{\alpha
p^{\prime}}&\text{if }\beta p^{\prime}>\gamma,\\\ N_{L}^{\gamma}+N_{L}^{\alpha
p^{\prime}}L^{p^{\prime}+1}&\text{if }\beta p^{\prime}=\gamma,\\\
N_{L}^{\gamma}+N_{L}^{\alpha p^{\prime}+\gamma-\beta p^{\prime}}&\text{if
}\beta p^{\prime}<\gamma,\end{cases}$

in terms of the accuracy $\epsilon$ behaves as follows,

$\displaystyle\mathcal{C}^{k,\sf
ML}_{q,\varepsilon_{s}}(X)\lesssim_{(\alpha,\beta,\gamma,A,p,q)}\begin{cases}\epsilon^{-\frac{\gamma}{\alpha}}+\epsilon^{-p^{\prime}}&\text{if
}\beta p^{\prime}>\gamma,\\\
\epsilon^{-\frac{\gamma}{\alpha}}+\epsilon^{-p^{\prime}}|\log_{A}\epsilon|^{p^{\prime}+1}&\text{if
}\beta p^{\prime}=\gamma,\\\
\epsilon^{-\frac{\gamma}{\alpha}}+\epsilon^{-p^{\prime}-\frac{\gamma-\beta
p^{\prime}}{\alpha}}&\text{if }\beta p^{\prime}<\gamma,\end{cases}$

which completes the proof of the assertion. ∎

######  Remark 3.26 (Strong convergence implies ($\alpha$)).

Lemma 3.19 shows that under the stability condition ($\sf stab$), assumption
($\alpha$) is satisfied whenever there exists a constant
$\widetilde{C}_{\alpha}\in(0,\infty)$ such that
$\|X-X_{\ell}\|_{L_{k}(\Omega;E)}\leq\widetilde{C}_{\alpha}N_{\ell}^{-\alpha}$
holds for all $\ell\in\mathbb{N}$.

######  Remark 3.27 (Comparison with single-level Monte Carlo).

Under the assumptions ($\alpha$), ($\gamma$), ($\sf stab$) the single-level
Monte Carlo approach of Corollary 3.20 requires to choose the level $L$ and
the number of samples $M_{L}$ such that

$N_{L}\eqsim\epsilon^{-\nicefrac{{1}}{{\alpha}}}\qquad\text{and}\qquad
M_{L}\eqsim\epsilon^{-p^{\prime}}\\!,$

in order to achieve a target accuracy $\mathrm{err}^{k,\sf
SL}_{q,\varepsilon_{s}}(X)=\epsilon\in(0,\infty)$. Thus, the single-level
Monte Carlo method to estimate $\mathbb{M}^{k}_{\varepsilon}[X]$ causes
computational cost of the order

$\mathcal{C}^{k,\sf
SL}_{q,\varepsilon_{s}}(X)\eqsim\mathcal{C}_{L}M_{L}\lesssim
N_{L}^{\gamma}M_{L}\eqsim\epsilon^{-\frac{\gamma}{\alpha}-p^{\prime}}\\!.$

######  Remark 3.28 (Comparison with MLMC in Hilbert spaces).

In the case that $E$ is a Hilbert space, we have that $p=p^{\prime}=2$ and the
computational costs in ($\mathcal{C}$) coincide e.g. with those of [12,
Theorem 1] for the two cases when $\beta p^{\prime}\neq\gamma$. In the
critical case $\beta p^{\prime}=2\beta=\gamma$, we obtain an additional log-
factor $|\log_{A}\epsilon|$. This is due to the fact that we do not assume
independence across the levels $\ell\in\\{1,\ldots,L\\}$. Note that this
independence can be exploited only if  (i) $E$is a Hilbert space, and  (ii)
the error is measured in the $L_{2}$-norm with respect to
$(\Omega,\mathcal{A},\mathbb{P})$.

######  Remark 3.29 (Full injective tensor norm).

On the symmetric injective tensor product space
$\otimes^{k,s}_{\varepsilon_{s}}E$, the full and symmetric injective tensor
norms, $\|\,\cdot\,\|_{\varepsilon}$ and $\|\,\cdot\,\|_{\varepsilon_{s}}$,
are equivalent, see (2.5). Therefore, the results of Subsections 3.2 and 3.3
on convergence of Monte Carlo methods of standard (Theorem 3.16), single-level
(Corollary 3.20) and multilevel type (Theorems 3.24 and 3.25) hold also with
respect to the stronger norm on $L_{q}(\Omega;\otimes^{k}_{\varepsilon}E)$,
with the additional constant $\tfrac{k^{k}}{k!}$.

##  4\. Applications

In this section we illustrate the preceding, abstract theory by several
examples of stochastic equations, where the need for the presently developed
modifications of the standard Monte Carlo theory is entailed either by
problem-specific constraints on the choices of non-Hilbertian function spaces
for well-posedness or by the interest in error estimates in norms on Banach
spaces (such as Hölder norms).

Specifically, Subsections 4.1 and 4.2 are concerned with the $k$th moment MLMC
finite element convergence analysis for explicit, linear, second-order
elliptic PDEs with random forcing (in dimensions $d\in\\{2,3\\}$) or random
diffusion coefficient (for $d=1$), respectively. Here, the right-hand side is
assumed to be an element of (or taking values in) $L_{p}(\mathrm{D})$ for some
$p\in(1,\infty)$, where $\mathrm{D}\subset\mathbb{R}^{d}$ denotes the spatial
domain. To obtain well-posed problems, the case $p\in(1,2)$ necessitates
variational formulations on Banach spaces, whereas for $p\in(2,\infty)$ such
formulations may be advantageous to derive error estimates in Hölder norms via
Sobolev embeddings.

In Subsection 4.3 we discuss the MLMC approximation of higher-order moments
for vector-valued stochastic processes ${X\colon[0,T]\times\Omega\to E}$ in
tensor norms of Hölder spaces $C^{\delta}([0,T];E)$ for problem-specific
Hölder exponents $\delta\in[0,1)$. These results are applicable to many semi-
discrete or fully discrete numerical schemes for SDEs and stochastic PDEs and
we give some explicit examples.

###  4.1. Linear elliptic PDEs with random forcing

Let $\mathrm{D}\subset\mathbb{R}^{d}$ with $d\in\\{2,3\\}$ be an open,
bounded, polytopal Lipschitz domain (with closure $\overline{\mathrm{D}}$)
and, for $p\in[1,\infty]$ and $m\in\mathbb{N}$, let $L_{p}(\mathrm{D})$ and
$W^{m}_{p}(\mathrm{D})$ denote the standard Lebesgue and Sobolev spaces of
real-valued functions on $\mathrm{D}$.

We write ${\mathring{W}}^{1}_{p}(\mathrm{D})$ for the closure of
$C^{\infty}_{c}(\mathrm{D})$ (the space of smooth functions with compact
support inside $\mathrm{D}$) with respect to the norm on
$W^{1}_{p}(\mathrm{D})$, and $W^{-1}_{p}(\mathrm{D})$ for the dual space of
$\mathring{W}^{1}_{p^{\prime}}(\mathrm{D})$, where
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

####  4.1.1. Deterministic model problem

We assume given deterministic, continuous diffusion coefficients $a_{ij}\in
C^{0}(\overline{\mathrm{D}})$, $a_{ij}=a_{ji}$, $1\leq i,j\leq d$, which are
uniformly positive definite. Thus, there exist constants
$0<\underline{a}\leq\overline{a}<\infty$ such that, for all
$x\in\overline{\mathrm{D}}$,

$\forall\phi,\psi\in\mathbb{R}^{d}:\quad\sum\limits_{i,j=1}^{d}a_{ij}(x)\phi_{i}\phi_{j}\geq\underline{a}\,\|\phi\|_{\mathbb{R}^{d}}^{2},\;\;\sum\limits_{i,j=1}^{d}a_{ij}(x)\phi_{i}\psi_{j}\leq\overline{a}\,\|\phi\|_{\mathbb{R}^{d}}\|\psi\|_{\mathbb{R}^{d}},$
(4.1)

where $\|\,\cdot\,\|_{\mathbb{R}^{d}}$ denotes the Euclidean norm on
$\mathbb{R}^{d}$.

For $p\in(1,\infty)$ and a given source term $f$ in $L_{p}(\mathrm{D})$ (which
in the sequel shall be generalized to be random), we consider the following
variational formulation of a homogeneous Dirichlet boundary value problem:
Find

$u\in\mathring{W}^{1}_{p}(\mathrm{D}):\;\;B(u,v)=\langle
f,v\rangle\quad\forall v\in\mathring{W}^{1}_{p^{\prime}}(\mathrm{D}).$ (4.2)

Here, $\langle\,\cdot\,,\,\cdot\,\rangle$ denotes the
$W^{-1}_{p}(\mathrm{D})\times\mathring{W}^{1}_{p^{\prime}}(\mathrm{D})$
duality pairing, and the bilinear form $B$ is given by

$B\colon\mathring{W}^{1}_{p}(\mathrm{D})\times\mathring{W}^{1}_{p^{\prime}}(\mathrm{D})\to\mathbb{R},\qquad
B(w,v):=\int_{\mathrm{D}}\sum_{i,j=1}^{d}a_{ij}(x)\tfrac{\partial}{\partial
x_{i}}w(x)\tfrac{\partial}{\partial x_{j}}v(x)\,\mathrm{d}x.$

Evidently, Hölder’s inequality implies continuity of $B$ on
$\mathring{W}^{1}_{p}(\mathrm{D})\times\mathring{W}^{1}_{p^{\prime}}(\mathrm{D})$.
However, as opposed to the Hilbert space case $p=p^{\prime}=2$, the uniform
strong ellipticity assumption in (4.1) is in general not sufficient to
guarantee that the mapping $\mathring{W}^{1}_{p}(\mathrm{D})\ni u\mapsto
B(u,\,\cdot\,)\in W^{-1}_{p}(\mathrm{D})$ is an isomorphism. For the case of
the Laplace operator (i.e., $a_{ij}(x)=\delta_{ij}$) and every
$p\in(1,\infty)$, an inf-sup condition and hence well-posedness of (4.2) have
been shown in [55, Theorem 6.1], see also [8, Equation (8.6.5)]. Following the
arguments used in [8, Section 8.6] this can be generalized to diffusion
coefficients $(a_{ij})_{i,j=1}^{d}$ satisfying (4.1), provided that $p$ is
sufficiently close to $2$. In what follows, we will _require_ for an
appropriate range of integrability indices $p\in(1,\infty)$ that (4.2) has a
unique solution and, moreover, that this solution is
$W^{2}_{p}(\mathrm{D})$-regular. This is summarized in the next assumption.

######  Assumption 4.1.

There exists $p_{0}\in(d,\infty)$ such that, for all $p\in(1,p_{0})$ and every
$f\in L_{p}(\mathrm{D})$, the variational problem (4.2) admits a unique
solution $u\in\mathring{W}^{1}_{p}(\mathrm{D})$, and

$\forall p\in(1,p_{0})\quad\exists C_{p}\in(0,\infty)\quad\forall f\in
L_{p}(\mathrm{D}):\quad\|u\|_{W^{2}_{p}(\mathrm{D})}\leq
C_{p}\|f\|_{L_{p}(\mathrm{D})}.$ (4.3)

Sufficient conditions for the $W^{2}_{p}(\mathrm{D})$-regularity (4.3) to hold
for the Laplace problem in polygons (i.e., $d=2$) can, for instance, be found
in [28, Theorem 4.3.2.4].

Since $p_{0}>d$ and $d\in\\{2,3\\}$ are assumed, for a given
$q\in[p_{0},\infty)$, we may choose ${p=\frac{qd}{d+q}\in(1,\min\\{d,q\\})}$
in (4.3) and conclude by continuity of the Sobolev embedding
$W^{2}_{p}(\mathrm{D})\subseteq W^{1}_{q}(\mathrm{D})$ and Hölder’s inequality
that
$\|u\|_{W^{1}_{q}(\mathrm{D})}\lesssim_{(q,\mathrm{D})}\|u\|_{W^{2}_{p}(\mathrm{D})}\lesssim_{p}\|f\|_{L_{p}(\mathrm{D})}\lesssim_{(q,\mathrm{D})}\|f\|_{L_{q}(\mathrm{D})}$.
For $q\in(1,p_{0})$ this estimate trivially holds by (4.3) so that we obtain
the following stability estimate for all $p\in(1,\infty)$:

$\forall
p\in(1,\infty):\quad\|u\|_{W^{1}_{p}(\mathrm{D})}\lesssim_{(p,\mathrm{D})}\|f\|_{L_{p}(\mathrm{D})}.$
(4.4)

####  4.1.2. Finite element approximation

For the numerical approximation, we use a conforming finite element method
(FEM) for (4.2) based on continuous, piecewise first-order Langrangean basis
functions on $\overline{\mathrm{D}}$: on a regular, simplicial triangulation
$\mathcal{T}_{h}$ of $\overline{\mathrm{D}}$ with mesh size $h\in(0,\infty)$,
we consider the finite-dimensional space

$S^{1}_{0}(\mathrm{D};\mathcal{T}_{h}):=\bigl{\\{}v\in
C^{0}(\overline{\mathrm{D}}):v|_{\partial\mathrm{D}}=0,\;v|_{T}\in\mathbb{P}_{1}\;\forall
T\in\mathcal{T}_{h}\bigr{\\}},$

where $\mathbb{P}_{1}$ denotes the space of polynomials of degree at most one.
The corresponding Galerkin discretization of (4.2) reads: Find

$u_{h}\in S^{1}_{0}(\mathrm{D};\mathcal{T}_{h}):\;\;B(u_{h},v_{h})=\langle
f,v_{h}\rangle\quad\forall v_{h}\in S^{1}_{0}(\mathrm{D};\mathcal{T}_{h}).$
(4.5)

Evidently, this finite-dimensional problem is equivalent to a linear system of
equations, with matrix that is symmetric and, by (4.1), positive definite, so
that there exists a unique solution $u_{h}\in
S^{1}_{0}(\mathrm{D};\mathcal{T}_{h})$ of (4.5).

Under Assumption 4.1 and the additional condition that

$\forall i,j\in\\{1,\ldots,d\\}:\quad a_{ij}\in
W^{1}_{q}(\mathrm{D})\quad\text{for some}\quad\begin{cases}q>2&\text{if
}d=2,\\\ q\geq\frac{12}{5}&\text{if }d=3,\end{cases}$ (4.6)

it is shown in [8, Theorem 8.5.3] that, for all $p\in(1,\infty)$, the Galerkin
projection $u_{h}$ in (4.5) is bounded in $\mathring{W}^{1}_{p}(\mathrm{D})$:
There exists a mesh size $h_{0}\in(0,\infty)$ such that

$\forall
h\in(0,h_{0}):\quad\|u_{h}\|_{W^{1}_{p}(\mathrm{D})}\lesssim_{(p,\mathrm{D})}\|u\|_{W^{1}_{p}(\mathrm{D})}.$
(4.7)

Combining (4.4) and (4.7) implies stability of both the exact solution $u$ and
its approximation $u_{h}$: For all $p\in(1,\infty)$, there exist a mesh width
$h_{0}\in(0,\infty)$ and a constant $\widetilde{C}_{\sf stab}\in(0,\infty)$
such that

$\forall
h\in(0,h_{0}):\quad\max\bigl{\\{}\|u\|_{W^{1}_{p}(\mathrm{D})},\|u_{h}\|_{W^{1}_{p}(\mathrm{D})}\bigr{\\}}\leq\widetilde{C}_{\sf
stab}\|f\|_{L_{p}(\mathrm{D})}.$ (4.8)

Moreover, by [8, Equation (8.5.4)] $u_{h}$ is quasi-optimal in
$\mathring{W}^{1}_{p}(\mathrm{D})$ for all $p\in(1,\infty)$: There exist
$h_{0},C_{\sf opt}\in(0,\infty)$ (which may depend on $\mathrm{D}$ and $p$)
such that

$\forall h\in(0,h_{0}):\quad\|u-u_{h}\|_{W^{1}_{p}(\mathrm{D})}\leq C_{\sf
opt}\inf_{v_{h}\in
S^{1}_{0}(\mathrm{D};\mathcal{T}_{h})}\|u-v_{h}\|_{W^{1}_{p}(\mathrm{D})}.$

Therefore, under Assumption 4.1 and (4.6), for every quasi-uniform family of
triangulations $(\mathcal{T}_{h})_{h\in\mathcal{H}}$, standard approximation
properties of the corresponding finite element spaces
$S^{1}_{0}(\mathrm{D};\mathcal{T}_{h})$, $h\in\mathcal{H}\subseteq(0,\infty)$,
show that, for all $p\in(0,p_{0})$,

$\forall
h\in\mathcal{H}\cap(0,h_{0}):\quad\|u-u_{h}\|_{W^{1}_{p}(\mathrm{D})}\lesssim_{(p,\mathrm{D})}h\,\|u\|_{W^{2}_{p}(\mathrm{D})}\lesssim_{(p,\mathrm{D})}h\,\|f\|_{L_{p}(\mathrm{D})},$
(4.9)

where we also used the assumed regularity (4.3).

####  4.1.3. Random forcing and MLMC-FEM

Suppose the setting of the previous subsections. In particular, the
coefficients $(a_{ij})_{i,j=1}^{d}$ satisfy (4.1) and (4.6), and Assumption
4.1 holds for some $p_{0}\in(d,\infty)$. In this subsection we fix
$p\in(1,p_{0})$, which determines the spatial integrability of a given random
forcing. Random forcing in (4.2) amounts to assuming that the right-hand side
$f$ is an element of $L_{r}(\Omega;L_{p}(\mathrm{D}))$ for some suitable
integrability index ${r\in[1,\infty)}$ with respect to the probability space
$(\Omega,\mathcal{A},\mathbb{P})$, i.e., we seek a
$\mathring{W}^{1}_{p}(\mathrm{D})$-valued random variable $u$ such that

$B(u(\omega),v)=\langle f(\omega),v\rangle\quad\forall
v\in\mathring{W}^{1}_{p^{\prime}}(\mathrm{D}),\;\;\;\text{for almost all
}\omega\in\Omega.$ (4.10)

Under the above mentioned regularity requirements, see Assumption 4.1, we may
argue for almost all $\omega\in\Omega$ to establish the existence and
uniqueness of a (stochastic) solution $u\in
L_{r}(\Omega;\mathring{W}^{1}_{p}(\mathrm{D}))$ satisfying (4.10), with

$u\in
L_{r}(\Omega;W^{2}_{p}(\mathrm{D})),\qquad\|u\|_{L_{r}(\Omega;W^{2}_{p}(\mathrm{D}))}\lesssim_{(p,\mathrm{D})}\|f\|_{L_{r}(\Omega;L_{p}(\mathrm{D}))}.$

_Multilevel finite element discretizations_ of (4.10) will be based on the
discrete variational problem (4.5), considered $\mathbb{P}$-a.s. To this end,
we denote by $\\{\mathcal{T}_{\ell}\\}_{\ell\in\mathbb{N}}$ a nested sequence
of regular, simplicial triangulations $\mathcal{T}_{\ell}$ of
$\overline{\mathrm{D}}$, with corresponding sequence of mesh sizes
$\\{h_{\ell}\\}_{\ell\in\mathbb{N}}$. We assume that $\mathcal{T}_{\ell+1}$ is
obtained from $\mathcal{T}_{\ell}$ via uniform red refinement. Then,
$h_{\ell+1}\lesssim h_{\ell}/2$ and, without loss of generality, we may assume
that $h_{1}<h_{0}$, with $h_{0}$ as in (4.7)–(4.9). The corresponding sequence
of Galerkin solutions $u_{h_{\ell}}\in
S^{1}_{0}(\mathrm{D};\mathcal{T}_{\ell})$ shall be denoted by $u_{\ell}$ (with
slight abuse of notation).

In the next corollary we verify that all assumptions of the
“$\alpha\beta\gamma$ theorem”, see Theorem 3.25, are satisfied to bound the
computational costs of the MLMC-FEM estimator for
$\mathbb{M}^{k}_{\varepsilon}[u]$ for a given accuracy and provide an upper
bound for these costs.

######  Corollary 4.2.

Let (4.1) and (4.6) be satisfied and suppose that Assumption 4.1 holds for
some $p_{0}\in(d,\infty)$. Assume that $p\in(1,p_{0})$,
$q\in[\min\\{p,2\\},\infty)$, and $k\in\mathbb{N}$. For $f\in
L_{kq}(\Omega;L_{p}(\mathrm{D}))$, let $u\in
L_{kq}(\Omega;\mathring{W}^{1}_{p}(\mathrm{D}))$ be the solution to (4.10).
Furthermore, let the FEM approximations $(u_{\ell})_{\ell\in\mathbb{N}}$ be
constructed as described above. Then, for $E:=W^{1}_{p}(\mathrm{D})$ and
$N_{\ell}:=\dim\bigl{(}S_{0}^{1}(\mathrm{D};\mathcal{T}_{\ell})\bigr{)}\eqsim
h_{\ell}^{-d}\eqsim A^{\ell}$, with $A:=2^{d}$, all conditions of Theorem
3.25 are fulfilled,

($\alpha$) $\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\bigl{\|}\mathbb{M}^{k}_{\varepsilon}[u]-\mathbb{M}^{k}_{\varepsilon}[u_{\ell}]\bigr{\|}_{\varepsilon_{s}}$
$\displaystyle\lesssim_{(k,p,\mathrm{D},f)}N_{\ell}^{-\nicefrac{{1}}{{d}}},$
$\displaystyle\text{i.e., }\alpha$ $\displaystyle=d^{-1},$ ($\beta$)
$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\|u_{\ell}-u_{\ell-1}\|_{L_{kq}(\Omega;W^{1}_{p}(\mathrm{D}))}$
$\displaystyle\lesssim_{(k,p,q,\mathrm{D},f)}N_{\ell}^{-\nicefrac{{1}}{{d}}},$
$\displaystyle\text{i.e., }\beta$ $\displaystyle=d^{-1},$ ($\gamma$)
$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\mathcal{C}_{\ell}$ $\displaystyle\lesssim N_{\ell}^{k},$
$\displaystyle\text{i.e., }\gamma$ $\displaystyle=k,$ ($\sf stab$)
$\displaystyle\quad\forall\ell\in\mathbb{N}:$
$\displaystyle\max\bigl{\\{}\|u\|_{L_{k}(\Omega;W^{1}_{p}(\mathrm{D}))},\|u_{\ell}$
$\displaystyle\|_{L_{kq}(\Omega;W^{1}_{p}(\mathrm{D}))}\bigr{\\}}\leq C_{\sf
stab},$

for some constant $C_{\sf stab}\in(0,\infty)$ depending only on $p,\mathrm{D}$
and $\|f\|_{L_{kq}(\Omega;L_{p}(\mathrm{D}))}$.

Furthermore, the $L_{q}$-accuracy $\mathrm{err}_{q,\varepsilon_{s}}^{k,\sf
ML}(u)<\epsilon\in(0,\nicefrac{{1}}{{2}}]$ of the multilevel Monte Carlo
estimator for $\mathbb{M}^{k}_{\varepsilon}[u]$ can be achieved at
computational costs of the order

$\mathcal{C}^{k,\sf
ML}_{q,\varepsilon_{s}}(u)\lesssim_{(k,p,q,\mathrm{D},f)}\begin{cases}\epsilon^{-\bar{p}^{\prime}}&\text{if}\quad\bar{p}^{\prime}>kd,\\\
\epsilon^{-kd}\,|\log_{2}\epsilon|^{kd+1}&\text{if}\quad\bar{p}^{\prime}=kd,\\\
\epsilon^{-kd}&\text{if}\quad\bar{p}^{\prime}<kd,\end{cases}$ (4.11)

where $\bar{p}^{\prime}\in[2,\infty)$ is such that
$\tfrac{1}{\min\\{p,2\\}}+\tfrac{1}{\bar{p}^{\prime}}=1$.

###### Proof.

First note that we may apply the deterministic stability estimate (4.8) for
almost all $\omega\in\Omega$, showing that, for all $\ell\in\mathbb{N}$,

$\max\bigl{\\{}\|u\|_{L_{k}(\Omega;W^{1}_{p}(\mathrm{D}))},\|u_{\ell}\|_{L_{kq}(\Omega;W^{1}_{p}(\mathrm{D}))}\bigr{\\}}\leq\widetilde{C}_{\sf
stab}\|f\|_{L_{kq}(\Omega;L_{p}(\mathrm{D}))}=:C_{\sf stab}.$

The integrability of $f\in L_{kq}(\Omega;L_{p}(\mathrm{D}))$ combined with the
deterministic FEM convergence result (4.9) implies strong convergence,

$\forall\ell\in\mathbb{N}:\quad\|u-u_{\ell}\|_{L_{kq}(\Omega;W^{1}_{p}(\mathrm{D}))}\lesssim_{(p,\mathrm{D})}h_{\ell}\,\|f\|_{L_{kq}(\Omega;L_{p}(\mathrm{D}))}.$

Since $h_{\ell}\eqsim N_{\ell}^{-\nicefrac{{1}}{{d}}}$, we conclude that the
conditions ($\alpha$) and ($\beta$) of Theorem 3.25 are satisfied with
$\alpha=\beta=d^{-1}$, where we also have used Remark 3.26 for ($\alpha$) and
the triangle inequality for ($\beta$).

Assuming a linear complexity solver (as, e.g., multigrid), the cost
$\mathcal{C}_{\ell,1}$ for computing one sample of $u_{\ell}$ in (4.5) is
bounded by $\mathcal{C}_{\ell,1}\leq C_{\gamma,1}N_{\ell}$ with some constant
$C_{\gamma,1}\in(0,\infty)$ independent of $\ell$. Since the computation of
the $k$th Kronecker product of a vector of length $N_{\ell}$ causes
computational cost of the magnitude $N_{\ell}^{k}$, the total cost
$\mathcal{C}_{\ell}$ for computing one sample of the random variable
$\xi_{\ell}=\otimes^{k}u_{\ell}-\otimes^{k}u_{\ell-1}$ is of the order
$N_{\ell}^{k}$, and condition ($\gamma$) is satisfied with $\gamma=k$.

Thus, the assumptions ($\alpha$), ($\beta$), ($\gamma$) and ($\sf stab$) of
Theorem 3.25 are satisfied, and the upper bound for the computational costs in
(4.11) follows upon applying Theorem 3.25, since the Banach space
$E=W^{1}_{p}(\mathrm{D})$ has type $\min\\{p,2\\}$. ∎

######  Remark 4.3.

In the Hilbert space case, it is in general not optimal to obtain a
convergence rate bound for ($\alpha$) by combining strong convergence with
stability ($\sf stab$), as outlined in Remark 3.26. For instance, the error
analysis of Galerkin approximations for generalized Whittle–Matérn fields in
[15, Proposition 4] reveals that the corresponding approximations of the
covariance function converge more than twice as fast in the
$L_{2}(\mathrm{D}{\times}\mathrm{D})$-norm as the corresponding Gaussian
random field approximations in the strong
$L_{q}(\Omega;L_{2}(\mathrm{D}))$-sense. However, it is not obvious if and how
this behavior generalizes to random variables with values in Banach spaces.

###  4.2. Linear elliptic PDEs with log-Gaussian coefficient

We next consider a linear, second-order elliptic PDE with mixed
Dirichlet–Neumann boundary conditions and right-hand side in
$L_{p}(\mathrm{D})$. As opposed to Subsection 4.1, we now assume that the
diffusion coefficient $a$ is random. More specifically, as e.g. in [9, 10,
56], we suppose that $a$ is log-Gaussian, i.e., $a(x)=\exp(g(x))$ for almost
all $x\in\mathrm{D}$, $\mathbb{P}$-a.s., for some Gaussian random field
$g\colon\mathrm{D}\times\Omega\to\mathbb{R}$. The spatial domain is assumed to
be a bounded interval $\mathrm{D}=I=(0,b)$ of length $b\in(0,\infty)$. The
restriction of the spatial dimension to $d=1$ facilitates an explicit
expression of the inf-sup constant of the bilinear form, appearing in the
corresponding variational formulation, depending on
$\underline{a}:=\operatorname{ess}\inf_{x\in I}a(x)$. In this setting,
$\underline{a}$ is a random variable satisfying $\underline{a}\in
L_{q}(\Omega;\mathbb{R})$ for all $q\in[1,\infty)$. Similarly as in [9], this,
in turn, yields well-posedness of the variational problem and strong
convergence of finite element approximations.

####  4.2.1. Deterministic model problem

For $p\in[1,\infty]$, $m\in\mathbb{N}$, we recall the Lebesgue and Sobolev
spaces $L_{p}(I)$ and $W^{m}_{p}(I)$ from Subsection 4.1. We furthermore note
that, since $d=1$, for every $p\in[1,\infty]$, elements $v$ in $W^{1}_{p}(I)$
coincide (upon a modification on a subset of $\overline{I}$ of zero Lebesgue
measure) with a unique function which is continuous on $\overline{I}=[0,b]$,
denoted by $\widetilde{v}\in C^{0}(\overline{I})$, and we define the subspace

$\mathring{W}_{p,\\{0\\}}^{1}(I):=\bigl{\\{}v\in
W^{1}_{p}(I):\widetilde{v}(0)=0\bigr{\\}}.$

In virtue of the Poincaré inequality, on this subspace the map
$v\mapsto|v|_{W^{1}_{p}(I)}$, $|v|_{W^{1}_{p}(I)}:=\|v^{\prime}\|_{L_{p}(I)}$
defines a norm, where $v^{\prime}$ denotes the weak derivative of $v$. In
addition, we write $p^{\prime}\in[1,\infty]$ for the Hölder conjugate of
$p\in[1,\infty]$, and we let $W^{-1}_{p,\\{0\\}}(I)$ be the dual space of
$\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I)$, equipped with the norm

$\|f\|_{W^{-1}_{p,\\{0\\}}(I)}:=\sup_{0\neq
v\in\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I)}\frac{\langle
f,v\rangle}{|v|_{W^{1}_{p^{\prime}}(I)}}.$

We assume given a finite partition
$\mathcal{P}=\\{J_{i}\\}_{i=1}^{n_{\mathcal{P}}}$ of pairwise disjoint, open
subintervals $J_{i}$ of $I$ such that
$\overline{J_{1}}\cup\ldots\cup\overline{J_{n_{\mathcal{P}}}}=\overline{I}=[0,b]$.
Furthermore, we suppose that the scalar diffusion coefficient satisfies

$a\in W^{1}_{\infty}(I;\mathcal{P}),\quad\text{where}\quad
W^{1}_{\infty}(I;\mathcal{P}):=\bigl{\\{}a\in L_{\infty}(I)\,|\,\forall
J\in\mathcal{P}:\,a|_{J}\in W^{1}_{\infty}(J)\bigr{\\}},$

and it is positive in the sense that there exist constants $\underline{a}$,
$\overline{a}$ such that

$0<\underline{a}\leq a(x)\leq\overline{a}<\infty\qquad\text{for almost all
}x\in I.$

For $p\in(1,\infty)$ and $f\in L_{p}(I)$ we then consider the following
boundary value problem, with mixed (Dirichlet–Neumann) boundary conditions:
Find

$u\colon\overline{I}\to\mathbb{R}:\quad\;-(a(x)u^{\prime}(x))^{\prime}=f(x),\quad\text{a.a.
}x\in I,\quad\;\;u(0)=a(b)u^{\prime}(b)=0.$ (4.12)

The weak formulation of (4.12) reads: Find

$u\in\mathring{W}^{1}_{p,\\{0\\}}(I):\;\;B_{a}(u,v)=\langle
f,v\rangle\quad\forall v\in\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I),$
(4.13)

where the bilinear form $B_{a}$ is defined by
$B_{a}(w,v):=\int_{0}^{b}a(x)w^{\prime}(x)v^{\prime}(x)\,\mathrm{d}x$, for all
$w\in\mathring{W}^{1}_{p,\\{0\\}}(I)$ and
$v\in\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I)$. For every $p\in(1,\infty)$,
existence and uniqueness of a solution $u$ to (4.13) follow from continuity of
$B_{a}$,

$\forall w\in\mathring{W}^{1}_{p,\\{0\\}}(I)\;\;\;\forall
v\in\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I):\quad|B_{a}(w,v)|\leq\overline{a}\,|w|_{W^{1}_{p}(I)}|v|_{W^{1}_{p^{\prime}}(I)},$
(4.14)

and the following _inf-sup condition_ :

$\inf_{0\neq w\in\mathring{W}^{1}_{p,\\{0\\}}(I)}\sup_{0\neq
v\in\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I)}\frac{B_{a}(w,v)}{|w|_{W^{1}_{p}(I)}|v|_{W^{1}_{p^{\prime}}(I)}}\geq\underline{a}.$
(4.15)

For the homogeneous Dirichlet boundary value problem ($u(0)=u(b)=0$), a
constructive proof for the inf-sup condition on
$\mathring{W}_{p}^{1}(I)\times\mathring{W}^{1}_{p^{\prime}}(I)$ has been given
in [1, Proof of Theorem 3.1]. We adjust the argument from [1], to derive
(4.15) for the problem (4.12) with mixed boundary conditions. To this end, let
$w\in\mathring{W}^{1}_{p,\\{0\\}}(I)\setminus\\{0\\}$ be arbitrary but fixed,
and define

$v_{w}(x):=\int_{0}^{x}\operatorname{sign}(w^{\prime}(t))|w^{\prime}(t)|^{p-1}\,\mathrm{d}t,\quad
x\in\overline{I}=[0,b].$ (4.16)

This function satisfies $v_{w}(0)=0$, and it is weakly differentiable with
weak derivative

$v_{w}^{\prime}(x)=\operatorname{sign}(w^{\prime}(x))|w^{\prime}(x)|^{p-1}\qquad\text{for
almost all }x\in I.$

We furthermore obtain that, for almost all $x\in I$,
$|v_{w}^{\prime}(x)|=|w^{\prime}(x)|^{p-1}=|w^{\prime}(x)|^{p/p^{\prime}}$,
and conclude that $v_{w}\in\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I)$, with

$|v_{w}|_{W^{1}_{p^{\prime}}(I)}=\|v_{w}^{\prime}\|_{L_{p^{\prime}}(I)}=\|w^{\prime}\|_{L_{p}(I)}^{p/p^{\prime}}=|w|_{W^{1}_{p}(I)}^{p/p^{\prime}}=|w|_{W^{1}_{p}(I)}^{p-1}.$

The continuity (4.14) of $B_{a}$ implies that $B_{a}(w,v_{w})$ is finite, and
we find that

$B_{a}(w,v_{w})=\int_{0}^{b}a(x)|w^{\prime}(x)|^{p}\,\mathrm{d}x\geq\underline{a}\,|w|_{W^{1}_{p}(I)}^{p}=\underline{a}\,|w|_{W^{1}_{p}(I)}|v_{w}|_{W^{1}_{p^{\prime}}(I)}.$

Since $w\in\mathring{W}^{1}_{p,\\{0\\}}(I)\setminus\\{0\\}$ was arbitrary,
(4.15) follows.

The inf-sup condition (4.15) (together with its symmetric counterpart which
can be shown in the same fashion) implies that, for every $f\in
W^{-1}_{p,\\{0\\}}(I)$, the variational problem (4.13) admits a unique
solution $u\in\mathring{W}^{1}_{p,\\{0\\}}(I)$. Furthermore, the linear data-
to-solution mapping $W^{-1}_{p,\\{0\\}}(I)\ni f\mapsto
u\in\mathring{W}^{1}_{p,\\{0\\}}(I)$ is an isomorphism with

$\|u^{\prime}\|_{L_{p}(I)}=|u|_{W^{1}_{p}(I)}\leq\underline{a}^{-1}\|f\|_{W^{-1}_{p,\\{0\\}}(I)}.$
(4.17)

In the case that $f\in L_{p}(I)$, this solution is more regular: Considering
the differential equation (4.12) in weak sense on $J\in\mathcal{P}$ implies,
for $a\in W^{1}_{\infty}(I;\mathcal{P})$ and $f\in L_{p}(I)$, that the second
weak derivative of $u|_{J}$ restricted to $J\subseteq I$ exists and

$-u|_{J}^{\prime\prime}(x)=a(x)^{-1}\left[f(x)+a|_{J}^{\prime}(x)u^{\prime}(x)\right]\quad\text{for
almost all }x\in J.$

Taking here the $L_{p}(J)$-norm yields with elementary estimates that, for
every $J\in\mathcal{P}$,

$\displaystyle\|u|_{J}^{\prime\prime}\|_{L_{p}(J)}$
$\displaystyle\leq\|a^{-1}\|_{L_{\infty}(J)}\left[\|f\|_{L_{p}(J)}+\|a|_{J}^{\prime}\|_{L_{\infty}(J)}\|u^{\prime}\|_{L_{p}(J)}\right]$
$\displaystyle\leq\underline{a}^{-1}\Bigl{[}\|f\|_{L_{p}(I)}+\underline{a}^{-1}\|a|_{J}^{\prime}\|_{L_{\infty}(J)}\|f\|_{W^{-1}_{p,\\{0\\}}(I)}\Bigr{]}\leq
C^{\sf\,reg}_{a,p}\,\|f\|_{L_{p}(I)},$

where the constant $C^{\sf\,reg}_{a,p}\in(0,\infty)$ is given by

$C^{\sf\,reg}_{a,p}:=\underline{a}^{-1}\Bigl{[}1+\underline{a}^{-1}\max_{J\in\mathcal{P}}\|a|_{J}^{\prime}\|_{L_{\infty}(J)}\,C_{L_{p}\to
W^{-1}_{p,\\{0\\}}}\Bigr{]},$

and $C_{L_{p}\to W^{-1}_{p,\\{0\\}}}:=\sup\nolimits_{f\in
B_{L_{p}(I)}}\|f\|_{W^{-1}_{p,\\{0\\}}(I)}$ denotes the norm of the continuous
embedding $L_{p}(I)\subset W^{-1}_{p,\\{0\\}}(I)$.

Hence, for every $f\in L_{p}(I)$, the unique weak solution to (4.13) satisfies

$u\in
W^{2}_{p}(I;\mathcal{P})\cap\mathring{W}^{1}_{p,\\{0\\}}(I),\qquad\max_{J\in\mathcal{P}}\|u|_{J}^{\prime\prime}\|_{L_{p}(J)}\leq
C^{\sf\,reg}_{a,p}\,\|f\|_{L_{p}(I)},$ (4.18)

where $W^{2}_{p}(I;\mathcal{P}):=\bigl{\\{}v\in W^{1}_{p}(I)\,|\,\forall
J\in\mathcal{P}:\,v|_{J}\in W^{2}_{p}(J)\bigr{\\}}$ is the space of functions
in $W^{1}_{p}(I)$ which are piecewise in $W^{2}_{p}$ on the partition
$\mathcal{P}$ of $I$.

####  4.2.2. Finite element approximation

For the numerical approximation of the solution
$u\in\mathring{W}^{1}_{p,\\{0\\}}(I)$ to (4.13) we use a similar conforming
finite element discretization as in Subsection 4.1.2. That is, we use
continuous, piecewise affine-linear functions on a partition $\mathcal{T}_{h}$
of $\overline{I}$ with mesh size $h\in(0,\infty)$,

$S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h}):=\bigl{\\{}v\in
C^{0}(\overline{I}):v(0)=0,\;v|_{T}\in\mathbb{P}_{1}\;\forall
T\in\mathcal{T}_{h}\bigr{\\}}.$

Evidently,
$S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})\subset\mathring{W}^{1}_{p,\\{0\\}}(I)\cap\mathring{W}^{1}_{p^{\prime}\\!,\\{0\\}}(I)$
and
$\dim\bigl{(}S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})\bigr{)}=\\#(\mathcal{T}_{h})$.
For given ${f\in L_{p}(I)}$, the Galerkin discretization of (4.13) reads: Find

$u_{h}\in S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h}):\;\;B_{a}(u_{h},v_{h})=\langle
f,v_{h}\rangle\quad\forall v_{h}\in S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h}).$
(4.19)

Unique solvability of (4.19) follows from the ($h$-uniform) discrete inf-sup
condition:

$\inf_{0\neq w_{h}\in S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})}\sup_{0\neq
v_{h}\in
S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})}\frac{B_{a}(w_{h},v_{h})}{|w_{h}|_{W^{1}_{p}(I)}|v_{h}|_{W^{1}_{p^{\prime}}(I)}}\geq\underline{a}.$
(4.20)

To verify (4.20), note that the proof of (4.15) carries over to the discrete
case: Given $w_{h}\in S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})$, one checks that
the expression (4.16) yields an element $v_{h}$ in
$S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})$, and that all steps in the proof of
(4.15) may be repeated verbatim.

The discrete inf-sup condition (4.20) and the continuity (4.14) imply that
(4.19) admits a unique solution $u_{h}\in
S^{1}_{0,\\{0\\}}(I;\mathcal{T}_{h})$ with

$\|u_{h}^{\prime}\|_{L_{p}(I)}=|u_{h}|_{W^{1}_{p}(I)}\leq\underline{a}^{-1}\|f\|_{W^{-1}_{p,\\{0\\}}(I)},$
(4.21)

which is, furthermore, quasi-optimal: