# Long-range dependent completely correlated mixed fractional Brownian motion

Josephine Dufitinema  Department of Mathematics and Statistics, University of
Vaasa, P.O. Box 700, FIN-65101 Vaasa, FINLAND<EMAIL_ADDRESS>,
Foad Shokrollahi  Department of Mathematics and Statistics, University of
Vaasa, P.O. Box 700, FIN-65101 Vaasa, FINLAND<EMAIL_ADDRESS>,
Tommi Sottinen  Department of Mathematics and Statistics, University of Vaasa,
P.O. Box 700, FIN-65101 Vaasa, FINLAND<EMAIL_ADDRESS>and  Lauri
Viitasaari  Department of Mathematics, Uppsala University, Box 480, 751 06
Uppsala, SWEDEN<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this paper we introduce the long-range dependent completely correlated
mixed fractional Brownian motion (ccmfBm). This is a process that is driven by
a mixture of Brownian motion (Bm) and a long-range dependent completely
correlated fractional Brownian motion (fBm, ccfBm) that is constructed from
the Brownian motion via the Molchan–Golosov representation. Thus, there is a
single Bm driving the mixed process. In the short time-scales the ccmfBm
behaves like the Bm (it has Brownian Hölder index and quadratic variation).
However, in the long time-scales it behaves like the fBm (it has long-range
dependence governed by the fBm’s Hurst index). We provide a transfer principle
for the ccmfBm and use it to construct the Cameron–Martin–Girsanov–Hitsuda
theorem and prediction formulas. Finally, we illustrate the ccmfBm by
simulations.

###### Key words and phrases:

Cameron–Martin–Girsanov theorem, fractional Brownian motion, fractional
Gaussian noises, long-range dependence, prediction, transfer principle

###### 2020 Mathematics Subject Classification:

Primary 60G22; Secondary 60G25, 60G15

##  1\. Introduction

Long range dependence have numerous applications in various models and have
been a topic of active research, see e.g. monographs [Beran-et-al-2013,
Samorodnitsky-2016] and references therein.

The fractional Brownian motion (fBm) is maybe the simplest model for long-
range dependence. Indeed, fBm is Gaussian, has stationary increments, and
continuous sample paths almost surely. For details on the fBm and it’s usage
in several applications, we refer to [12]. On the other hand, fBm is not a
suitable model if one requires long-range dependence in long time scales and,
at the same time, somehow different behaviour in the short time scales. For
example in finance, empirical studies suggest long-range dependence while the
standard Brownian motion (Bm) would be more suitable model in the short time
scales. This can be modelled by a so-called mixed fractional Brownian motion
(mfBm) that inherits short time behaviour from the Brownian motion and long
time behaviour from the fBm. Cheridito [6] introduced an independent mfBm as a
sum of a Brownian motion and an independent fBm that has received attention in
the literature ever since.

One of the wanted features for a Gaussian model is the so-called transfer
principle that allows to reconstruct underlying Brownian motion from the
observations. Indeed, once the transfer principle is established, several
applications including prediction formulas and parameter estimation become
rather straightforward. For this reason transfer principles are actively
studied in the literature, see e.g. [5] in the case of independent mfBm and
[20] for the general case in the abstract setting.

In this article we introduce the long-range dependent completely correlated
mixed fractional Brownian motion (ccmfBm). We study its basic properties and
provide a transfer principle for it. In comparison to the independent mfBm, in
our case the transfer principle is explicit allowing easily computable
formulas and model fitting in different applications.

The rest of the paper is organized as follows: In Section 2 we recall what is
the fractional Brownian motion (fBm) and recall its basic properties, and
introduce the long-range dependent completely correlated mixed fractional
Brownian motion (ccmfBm) and state its basic properties. In Section 3 we
develop the transfer principle. In Section 4 develop the
Cameron–Martin–Girsanov theorem for the ccmfBm by using the transfer
principle. As an application we show how to use the Cameron–Martin–Girsanov in
estimating a drift parameter in a ccmfBm model. In Section 5 we state the
prediction formula for the ccmfBm that follows directly from the transfer
principle. In Section 6 we illustrate the ccmfBm by simulations. Finally, in
Section 7 we summarize our findings and discuss the short-range dependent
ccmfBm shortly.

##  2\. Definitions and Basic Properties

We recall briefly what is the fractional Brownian motion and state its basic
properties. Then we introduce the long-range dependent completely correlated
mixed fractional Brownian motion and state its basic properties.

The fractional Brownian motion (fBm) $B^{H}={(B^{H}_{t})}_{t\geq 0}$ with
Hurst index $H\in(0,1)$, introduced by Kolmogorov [10] and christened by
Mandelbrot and Van Ness [11], is the centered Gaussian process having the
covariance function

$R_{H}(t,s)=\frac{1}{2}\left[t^{2H}+s^{2H}-|t-s|^{2H}\right].$

The fBm is the (up to a multiplicative constant) unique centered Gaussian
process that has stationary increments and is self-similar with the Hurst
index $H$. In particular, if $H=\frac{1}{2}$, then the fBm is the standard
Brownian motion (Bm). For more details on the fBm, we refer to Biagini et al.
[4] and Mishura [12]. In this article, we consider a generalization of the fBm
where the fBm is mixed with a Bm that generates the fBm. For other
generalizations of the fBm, see e.g. [1, 6, 14, 15].

For $H\in(\frac{1}{2},1)$ the fractional Brownian motion is long-range
dependent in the sense of the following “non-stationary” definition: A
stochastic process $Y$ is long-range dependent if, for all $t_{0}\geq 0$ and
lags $\delta>0$, the incremental covariance

$\rho_{Y}(t_{0},\delta;t)=\mathbb{E}\left[\left(Y_{t_{0}+\delta}-Y_{t_{0}}\right)\left(Y_{t+\delta}-Y_{t}\right)\right]$

decay at most following a power law:

$\rho_{Y}(t_{0},\delta;t)\geq\frac{C_{t_{0},\delta}}{t^{\beta}}$

for some $\beta>0$. Indeed, for fBm with $H>1/2$, we have

$\displaystyle\rho_{B^{H}}(t_{0},\delta;t)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(B^{H}_{t_{0}+\delta}-B^{H}_{t_{0}}\right)\left(B^{H}_{t+\delta}-B^{H}_{t}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\delta^{2H}}\mathbb{E}\left[B^{H}_{1}\left(B^{H}_{\frac{t-t_{0}}{\delta}+1}-B^{H}_{\frac{t-t_{0}}{\delta}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\delta^{2H}}\int_{0}^{1}\int_{\frac{t-t_{0}}{\delta}}^{\frac{t-t_{0}}{\delta}+1}\frac{\partial^{2}R_{H}}{\partial
u\partial v}(u,v)\,\mathrm{d}u\mathrm{d}v$ $\displaystyle=$
$\displaystyle\frac{H(2H-1)}{\delta^{2H}}\int_{0}^{1}\int_{\frac{t-t_{0}}{\delta}}^{\frac{t-t_{0}}{\delta}+1}(u-v)^{2H-2}\,\mathrm{d}u\mathrm{d}v$
$\displaystyle\sim$
$\displaystyle\frac{H(2H-1)\delta^{2}}{\left(t-t_{0}\right)^{2-2H}}.$

Here and in what follows, we use the notation $f(t)\sim g(t)$ for asymptotic
equivalence meaning that $\lim f(t)/g(t)=1$.

######  Remark 1.

We only consider the case $H\in(1/2,1)$, even though some of the results would
be true also for the short-range dependent case where $H\in(0,1/2)$. See
Section 7 for further discussion of the short-range completely correlated
mixed fractional Brownian motions.

Denote

$c(H)=\sqrt{\frac{2H\Gamma(\frac{3}{2}-H)}{\Gamma(H+\frac{1}{2})\Gamma(2-2H)}}\left(H-\frac{1}{2}\right)$

where $\Gamma$ is the Gamma function. Set

(1)
$K_{H}(t,s)=c(H)\frac{1}{s^{H-\frac{1}{2}}}\int_{s}^{t}\frac{u^{H-\frac{1}{2}}}{(u-s)^{\frac{3}{2}-H}}\,\mathrm{d}u.$

Denote $t\wedge s=\min(t,s)$. Then

$R_{H}(t,s)=\int_{0}^{{t\wedge s}}K_{H}(t,u)K_{H}(s,u)\,\mathrm{d}u,$

and consequently the following Molchan–Golosov [13] representation holds:

(2) $B^{H}_{t}=\int_{0}^{t}K_{H}(t,s)\,\mathrm{d}W_{s},$

where $W$ is a Bm. We note that the representation (2) is of Volterra type
meaning that $K_{H}(t,s)=0$ when $s>t$.

######  Definition 1.

The completely correlated mixed fractional Brownian motion (ccmfBm) is

$X=aW+bB^{H}$

where $a,b\in\mathbb{R}$ with $ab\neq 0$, $W$ is a Bm and $B^{H}$ is a fBm
constructed from $W$ via (2).

Unlike the mixed fBm with independent summands (see e.g. Cheridito [6]), the
ccmfBm does not have stationary increments. It however shares some path-
properties and long-range dependence with the independent summands fBm.

######  Proposition 1.

The ccmfBm is a centered Gaussian process with the covariance function

$\displaystyle R(t,s)$ $\displaystyle=$ $\displaystyle a^{2}({t\wedge
s})+ab\int_{0}^{{t\wedge
s}}\left[K_{H}(t,u)+K_{H}(s,u)\right]\,\mathrm{d}u+b^{2}R_{H}(t,s).$

Moreover, the ccmfBm

  1. (i)

is Hölder continuous with index $1/2$,

  2. (ii)

has quadratic variation $t\mapsto a^{2}t$,

  3. (iii)

is long-range dependent having the same power law decay in its autocovariance
as the fBm.

###### Proof.

It is clear that $X$ is centered. The covariance (1) follows from the Itô-
isometry

$\displaystyle\mathbb{E}\left[W_{t}B^{H}_{s}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\mathbf{1}_{t}(u)\,\mathrm{d}W_{u}\,\int_{0}^{T}K_{H}(s,u)\,\mathrm{d}W_{u}\right]$
$\displaystyle=$ $\displaystyle\int_{0}^{{t\wedge s}}K_{H}(s,u)\,\mathrm{d}u.$

(i) The Hölder index follows from Theorem 1 of [2] and (1). Indeed, let $s<t$.
Then

$\displaystyle\mathbb{E}\left[(X_{t}-X_{s})^{2}\right]$ $\displaystyle=$
$\displaystyle
a^{2}|t-s|+ab\mathbb{E}\left[\left(W_{t}-W_{s}\right)\left(B_{t}^{H}-B_{s}^{H}\right)\right]+b^{2}|t-s|^{2H}$

and, by Itô-isometry and Volterra property,

$\displaystyle\mathbb{E}\left[\left(W_{t}-W_{s}\right)\left(B_{t}^{H}-B_{s}^{H}\right)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\left(\mathbf{1}_{t}(u)-\mathbf{1}_{s}(u)\right)\left(K_{H}(t,u)-K_{H}(s,u)\right)\,\mathrm{d}u$
$\displaystyle=$ $\displaystyle\int_{s}^{t}K_{H}(t,u)\,\mathrm{d}u$
$\displaystyle=$ $\displaystyle K_{H}(t,u^{*})|t-s|$

for some $u^{*}\in[s,t]$ as $K_{H}(t,s)$ is continuous in $s$. Since
$K_{H}(t,s)\to 0$ as $s\to t$, we see that

$\mathbb{E}\left[(X_{t}-X_{s})^{2}\right]\sim a^{2}|t-s|,$

which shows that the Hölder index is $1/2$.

(ii) The quadratic variation comes from the fact that $B^{H}$ has zero
quadratic variation for $H\in(1/2,1)$ applied to Example 1 of [3] (see also
[7]).

(iii) Finally, let us show the long-range dependence. Now,

$\displaystyle\rho_{X}(t_{0},\delta;t)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left(X_{t_{0}+\delta}-X_{t_{0}}\right)\left(X_{t+\delta}-X_{t}\right)\right]$
$\displaystyle=$ $\displaystyle
a^{2}\mathbb{E}\left[\left(W_{t_{0}+\delta}-W_{t_{0}}\right)\left(W_{t+\delta}-W_{t}\right)\right]$
$\displaystyle+ab\mathbb{E}\left[\left(W_{t_{0}+\delta}-W_{t_{0}}\right)\left(B^{H}_{t+\delta}-B^{H}_{t}\right)\right]$
$\displaystyle+ab\mathbb{E}\left[\left(B^{H}_{t_{0}+\delta}-B^{H}_{t_{0}}\right)\left(W_{t+\delta}-W_{t}\right)\right]$
$\displaystyle+b^{2}\mathbb{E}\left[\left(B^{H}_{t_{0}+\delta}-B^{H}_{t_{0}}\right)\left(B^{H}_{t+\delta}-B^{H}_{t}\right)\right]$
$\displaystyle\sim$ $\displaystyle
ab\mathbb{E}\left[\left(W_{t_{0}+\delta}-W_{t_{0}}\right)\left(B^{H}_{t+\delta}-B^{H}_{t}\right)\right]$
$\displaystyle+b^{2}\rho_{B^{H}}(t_{0},\delta;t).$

The long-range dependence follows by noting that

$\displaystyle\mathbb{E}\left[\left(W_{t_{0}+\delta}-W_{t_{0}}\right)\left(B^{H}_{t+\delta}-B^{H}_{t}\right)\right]$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{0}+\delta}\left[K_{H}(t+\delta,u)-K_{H}(t,u)\right]\,\mathrm{d}u$
$\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{0}+\delta}\int_{t}^{t+\delta}\frac{\partial
K_{H}}{\partial v}(v,u)\,\mathrm{d}v\,\mathrm{d}u$ $\displaystyle\sim$
$\displaystyle\delta^{2}\frac{\partial K_{H}}{\partial t}(t,t_{0})$
$\displaystyle=$
$\displaystyle\delta^{2}c(H)\left(\frac{t}{t_{0}}\right)^{H-\frac{1}{2}}\frac{1}{(t-t_{0})^{\frac{3}{2}-H}},$

which is the same power decay law, namely $t^{2H-2}$, as the fBm part has:
$\rho_{B^{H}}(t_{0},\delta;t)$. ∎

Unlike the case of independent summands mfBm with $a=b$ and $H\in(3/4,1)$ (see
Cheridito [6]), the ccmfBm is not equivalent to a Bm in any range of
$H\in(1/2,1)$:

######  Proposition 2.

Let $b\neq 0$. Then the law of the ccmfBm on $[0,T]$ is singular to the law of
any multiple of Bm on $[0,T]$.

###### Proof.

If follows from Hitsuda [9] and the Itô-isometry that the ccmfBm is equivalent
to a multiple of Brownian motion if and only if

$\int_{0}^{T}\int_{0}^{T}\left[\frac{\partial K_{H}}{\partial
t}(t,s)\right]^{2}\,\mathrm{d}s\,\mathrm{d}t<\infty.$

But

$\displaystyle\int_{0}^{T}\int_{0}^{T}\left[\frac{\partial K_{H}}{\partial
t}(t,s)\right]^{2}\,\mathrm{d}s\,\mathrm{d}t$ $\displaystyle=$ $\displaystyle
c_{1}(\alpha)^{2}\int_{0}^{T}\int_{s}^{T}\left(\frac{t}{s}\right)^{2H-1}\frac{\mathrm{d}t}{(t-s)^{3-2H}}\,\mathrm{d}s$
$\displaystyle\geq$ $\displaystyle
c(H)^{2}\int_{0}^{T}\int_{s}^{T}\frac{\mathrm{d}t}{(t-s)^{3-2H}}\,\mathrm{d}s$
$\displaystyle=$ $\displaystyle
c(H)^{2}\int_{0}^{T}\int_{0}^{T-s}\frac{\mathrm{d}u}{u^{3-2H}}\,\mathrm{d}s$
$\displaystyle=$ $\displaystyle\infty$

due to the non-integrable singularity of $1/u^{3-2H}$ at $u=0$.

Since Gaussian laws are always either equivalent or singular, the claim
follows. ∎

##  3\. Transfer principle

The transfer principle states that from the ccmfBm we can construct a Bm in a
non-anticipative way (the inverse transfer principle) and then represent the
ccmfBm in a non-anticipative way by using the constructed Bm (the direct
transfer principle). We consider processes on a compact time interval $[0,T]$
in this section.

Let $L^{2}=L^{2}([0,T])$ and $\|\cdot\|_{2}=\|\cdot\|_{L^{2}}$.

For a kernel $K\colon[0,T]^{2}\to\mathbb{R}$ its associated operator is

$\mathrm{K}f(t)=\int_{0}^{T}f(s)K(t,s)\,\mathrm{d}u.$

The adjoint associated operator $\mathrm{K}^{*}$ of a kernel $K$ is defined by
linearly extending the relation

(4) $\mathrm{K}^{*}\mathbf{1}_{t}(s)=K(t,s),$

where $\mathbf{1}_{t}=\mathbf{1}_{[0,t)}$ is the indicator function.

Since the kernel $K_{H}(t,s)$ is differentiable in $t$ and $K_{H}(t,t-)=0$,
its adjoint associated operator can be written as

(5) $\mathrm{K}_{H}^{*}f(t)=\int_{t}^{T}f(u)\frac{\partial K_{H}}{\partial
u}(u,t)\,\mathrm{d}u.$

Indeed, to verify formula (5) it is enough to check that it satisfies the
relation (4).

Let $\Lambda$ be the closure of the indicator functions $\mathbf{1}_{t}$,
$t\in[0,T]$, under the inner product generated by the relation

${\langle\mathbf{1}_{t},\mathbf{1}_{s}\rangle}_{\Lambda}=R(t,s).$

Let $\mathscr{H}_{1}$ be the linear space, or first chaos, of $X$, i.e., the
closure of the random variables $X_{t}$, $t\in[0,T]$, in $L^{2}(\Omega)$.

For $f\in\Lambda$ the abstract Wiener integral

$\int_{0}^{T}f(t)\,\mathrm{d}X_{t}$

is the image of the isometry $\mathbf{1}_{t}\mapsto X_{t}$ from $\Lambda$ to
$\mathscr{H}_{1}$.

Denote $L(t,s)=a\mathbf{1}_{t}(s)+bK_{H}(t,s)$ and let $\mathrm{L}$ and
$\mathrm{L}^{*}$ be the associated and adjoint associated operators of $L$.

######  Lemma 1.

$\mathrm{L}^{*}$ is a bounded operator on $L^{2}$ and it can be represented as

$\displaystyle\mathrm{L}^{*}f(t)$ $\displaystyle=$ $\displaystyle
af(t)+b\int_{t}^{T}f(u)\frac{\partial K_{H}}{\partial u}(u,t)\,\mathrm{d}u$
$\displaystyle=$ $\displaystyle
af(t)+\frac{bc(H)}{t^{H-\frac{1}{2}}}\int_{t}^{T}f(u)\frac{u^{H-\frac{1}{2}}}{(u-t)^{\frac{3}{2}-H}}\,\mathrm{d}u.$

###### Proof.

Let us first note that $\mathrm{L}^{*}=a\mathrm{I}^{*}+b\mathrm{K}^{*}_{H}$,
where $\mathrm{I}^{*}$ is the identity operator and $\mathrm{K}_{H}^{*}$ is
the adjoint associated operator defined in (5), where $K_{H}(t,s)$ is the
Molchan–Golosov kernel $K_{H}$ defined in (1). Since

${\|\mathrm{L}^{*}f\|}_{2}\leq|a|{\|f\|}_{2}+|b|{\|\mathrm{K}^{*}_{H}f\|}_{2},$

$\mathrm{L}^{*}$ is bounded in $L^{2}$, if $\mathrm{K}^{*}_{H}$ is bounded in
$L^{2}$. Also, we note that (1) is true for step functions. So, if
$\mathrm{L}^{*}$ is bounded in $L^{2}$, the formula (1) extends to all
functions in $L^{2}$. Finally we note that $\mathrm{K}^{*}_{H}$ is bounded on
$L^{2}$, because (for step functions $f$)

$\displaystyle{\|\mathrm{K}^{*}_{H}f\|}_{2}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\left[\mathrm{K}_{H}^{*}f(t)\right]^{2}\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\int_{0}^{T}f(t)f(s)\frac{\partial^{2}R_{H}}{\partial
s\partial t}(t,s)\,\mathrm{d}s\mathrm{d}t$ $\displaystyle=$ $\displaystyle
H(2H-1)\int_{0}^{T}\int_{0}^{T}\frac{f(t)f(s)}{|t-s|^{2-2H}}\,\mathrm{d}s\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
H(2H-1)\int_{0}^{T}\int_{0}^{T}\frac{f(t)^{2}}{|t-s|^{2-2H}}\,\mathrm{d}s\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
H(2H-1)\frac{T^{2H-1}}{H-\frac{1}{2}}{\|f\|}^{2}_{2},$

where we have used the elementary estimate

$2|f(t)f(s)|\leq f(t)^{2}+f(s)^{2}$

and symmetry. ∎

######  Lemma 2.

For each $t\in[0,T]$, the integral equation

(7) $\mathbf{1}_{t}(s)=aL^{-1}(t,s)+b\int_{s}^{T}L^{-1}(t,u)\frac{\partial
K_{H}}{\partial u}(u,s)\,\mathrm{d}u$

admits the unique $L^{2}$-solution given by

(8)
$L^{-1}(t,s)=\frac{1}{a}\mathbf{1}_{t}(s)+\frac{1}{a}\sum_{k=1}^{\infty}(-1)^{k}\left(\frac{b}{a}\right)^{k}\gamma_{k}(t,s)$

where

$\gamma_{k}(t,s)=\frac{c(H)^{k}\Gamma(H-\frac{1}{2})^{k}}{\Gamma\left(k\left(H-\frac{1}{2}\right)\right)}\frac{1}{s^{H-\frac{1}{2}}}\int_{s}^{t}u^{H-\frac{1}{2}}(u-s)^{k(H-\frac{1}{2})-1}\,\mathrm{d}u.$

###### Proof.

Denote

$G(s,u)=-\frac{bc(H)}{a}\frac{u^{H-\frac{1}{2}}}{s^{H-\frac{1}{2}}(u-s)^{\frac{3}{2}-H}}.$

Then (7) is the anti-Volterra equation of the second kind

$\frac{1}{a}\mathbf{1}_{t}(s)=L^{-1}(t,s)-\int_{s}^{t}L^{-1}(t,u)G(s,u)\,\mathrm{d}u.$

Since $\mathrm{L}^{*}=a\mathrm{I}^{*}-a\mathrm{G}$, Lemma 1 implies that the
solution of the equation (7) is given by the $L^{2}$-convergent
Liouville–Neumann series

(9)
$L^{-1}(t,s)=\sum_{k=1}^{\infty}\mathrm{G}^{k}\left[\frac{1}{a}\mathbf{1}_{t}\right](s),$

where $\mathrm{G}^{0}$ is the identity operator and
$\mathrm{G}^{k+1}=\mathrm{G}\mathrm{G}^{k}$. Formula (8) follows from formula
(9) by induction by using the formula

$\int_{s}^{u}(v-s)^{k\alpha-1}(u-v)^{\alpha-1}\,\mathrm{d}v=\frac{\Gamma(k\alpha)\Gamma(\alpha)}{\Gamma((k+1)\alpha)}(u-s)^{(k+1)\alpha-1},$

where $\alpha=H-\frac{1}{2}$. ∎

######  Remark 2.

The series (8) converges fast. Indeed, by using the Stirling’s approximation

$\Gamma\left(k\left(H-\frac{1}{2}\right)\right)\sim\sqrt{2\pi}\left(k\left(H-\frac{1}{2}\right)\right)^{k\left(H-\frac{1}{2}\right)-\frac{1}{2}}\mathrm{e}^{-k\left(H-\frac{1}{2}\right)}$

and the estimate

$\displaystyle\frac{1}{s^{H-\frac{1}{2}}}\int_{s}^{t}u^{H-\frac{1}{2}}(u-s)^{k\left(H-\frac{1}{2}\right)-1}\,\mathrm{d}u$
$\displaystyle\leq$
$\displaystyle\left(\frac{t}{s}\right)^{H-\frac{1}{2}}\frac{1}{k\left(H-\frac{1}{2}\right)}(t-s)^{k\left(H-\frac{1}{2}\right)}$

we obtain

$\displaystyle\gamma_{k}(t,s)\leq\frac{C^{k}}{\left(k\left(H-\frac{1}{2}\right)\right)^{k\left(H-\frac{1}{2}\right)+\frac{1}{2}}}\left(\frac{t}{s}\right)^{H-\frac{1}{2}}(t-s)^{k\left(H-\frac{1}{2}\right)},$

which also shows that (8) converges uniformly for all $s\in[\varepsilon,T]$.

Figure 1. The summands $\gamma_{k}(t,s)$ for $k=1,2,3$ (dashed lines) and
$k=10,15$ (solid lines) with $t=1$, $a=b=1$ and $H=0.75$.

From Lemma 2 we obtain directly the following basic form of the transfer
principle that is the main result of this paper.

######  Theorem 1.

Let $L^{-1}(t,s)$ be given by (8). The ccmfBm $X$ is an invertible Gaussian
Volterra process in the sense that the process $W$ defined as the abstract
Wiener integral

$W_{t}=\int_{0}^{t}L^{-1}(t,s)\,\mathrm{d}X_{s}$

is a Bm, and the ccmfBm can be reconstructed from it by the Wiener integral

$X_{t}=\int_{0}^{t}L(t,s)\,\mathrm{d}W_{s},$

where

$L(t,s)=a\mathbf{1}_{t}(s)+bK_{H}(t,s)$

and $K_{H}(t,s)$ is the Molchan–Golosov kernel given by (1).

Figure 2. The kernels $L(t,s)$ for $t=1$, $a=b=1$ and $H=0.6$ (solid line),
$H=0.75$ (dotted line) and $H=0.9$ (dashed line). Figure 3. The kernels
$L^{-1}(t,s)$ for $t=1$, $a=b=1$ and $H=0.6$ (solid line), $H=0.75$ (dotted
line) and $H=0.9$ (dashed line).

By Lemma 1 and Lemma 2 we have $\Lambda=L^{2}$. Therefore, Theorem 1 extends
to a transfer principle for deterministic integrands on $L^{2}$ with respect
to a Bm and ccmfBm. Actually, we could extend the transfer principle to
stochastic integrands by using Malliavin calculus and Skorokhod integration as
explained in [19, 20]. However, we omit that extension here.

######  Theorem 2.

Let $f\in L^{2}$. Let $X$ and $W$ be connected by Theorem 1. Then

$\displaystyle\int_{0}^{T}f(t)\,\mathrm{d}X_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\mathrm{L}^{*}f(t)\,\mathrm{d}W_{t},$
$\displaystyle\int_{0}^{T}f(t)\,\mathrm{d}W_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}(\mathrm{L}^{*})^{-1}f(t)\,\mathrm{d}X_{t},$

where

$\displaystyle\mathrm{L}^{*}f(t)$ $\displaystyle=$ $\displaystyle
af(t)+b\int_{t}^{T}f(s)\frac{\partial K_{H}}{\partial s}(s,t)\,\mathrm{d}s,$
$\displaystyle(\mathrm{L}^{*})^{-1}f(t)$ $\displaystyle=$ $\displaystyle
f(t)L^{-1}(T,t)+\int_{t}^{T}\left[f(s)-f(t)\right]L^{-1}(\mathrm{d}s,t).$

###### Proof.

Using (8) we can verify that the kernel $L^{-1}(t,s)$ is of bounded variation
in $t$. Moreover, if $f$ is a step function, Theorem 2 follows from Theorem 1
by linearity and straightforward computations. The claim now follows since
$\Lambda=L^{2}$ and the operator $\mathrm{L}^{*}$ is an isometry on $L^{2}$
(see also [20] for details). ∎

######  Remark 3.

Unlike the independent mfBm introduced by Cheridito [6], the ccmfBm does not
have stationary increments. Hence, it could be argued that the independent
mfBm is more natural. However, the inverse transfer principle for the ccmfBm
has a rather explicit and convenient form. For the independent mfBm the
inverse transfer principle is implicit and involves solving integral equations
numerically, see Cai et al. [5]. Therefore, the ccmfBm is more convenient for
applications. We also note that the Hölder continuity properties, quadratic
variation, and the long-range dependence of the independent mfBm and ccmfBm
are the same.

##  4\. Cameron–Martin–Girsanov–Hitsuda theorem

In this section we show how the transfer principle of Theorem 2 can be used to
characterize Gaussian processes that are equivalent in law to the ccmfBm and
to provide the corresponding Cameron–Martin–Girsanov theorem in the same way
as in [16, 18]. In this section we consider process on a compact time interval
$[0,T]$.

By the Hitsuda representation theorem [9] a Gaussian process $\tilde{W}$ is
equivalent to a Bm $W$ if and only if it can be represented

(10)
$\tilde{W}_{t}=W_{t}-\int_{0}^{t}\int_{0}^{s}\ell(s,u)\,\mathrm{d}W_{u}\,\mathrm{d}s-\int_{0}^{t}g(s)\,\mathrm{d}s$

for some $\ell\in L^{2}([0,T]^{2})$ and $g\in L^{2}([0,T])$. Here $W$ is a Bm
that is constructed from $\tilde{W}$ by

(11)
$W_{t}=\tilde{W}_{t}-\int_{0}^{t}\int_{0}^{s}\tilde{\ell}(s,u)\,\mathrm{d}\tilde{W}_{u}\,\mathrm{d}s-\int_{0}^{t}\left[g(s)-\int_{0}^{u}\tilde{\ell}(s,u)\,\mathrm{d}u\right]\,\mathrm{d}s,$

where $\tilde{\ell}$, the resolvent of $\ell$, is given by

(12) $\tilde{\ell}(t,s)=\sum_{k=1}^{\infty}\ell^{k}(t,s),$

where

$\displaystyle\ell^{1}(t,s)$ $\displaystyle=$ $\displaystyle\ell(t,s),$
$\displaystyle\ell^{k+1}(t,s)$ $\displaystyle=$
$\displaystyle\int_{s}^{t}\ell(t,u)\ell^{k}(u,s)\,\mathrm{d}u.$

The likelihood ratio of $\tilde{W}$ over $W$ given observation
$\mathscr{F}_{t}$ on the interval $[0,t]$ is

$\displaystyle\varphi_{t}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{d}\tilde{\mathbb{P}}}{\mathrm{d}\mathbb{P}}\Big{|}\mathscr{F}_{t}$
$\displaystyle=$
$\displaystyle\exp\bigg{\\{}\int_{0}^{t}\left[\int_{0}^{s}\ell(s,u)\mathrm{d}W_{u}+g(s)\right]\mathrm{d}W_{s}$
$\displaystyle-\frac{1}{2}\int_{0}^{t}\left[\int_{0}^{s}\ell(s,u)\mathrm{d}W_{u}-g(s)\right]^{2}\mathrm{d}s\bigg{\\}}$
$\displaystyle=$
$\displaystyle\exp\bigg{\\{}\int_{0}^{t}\left[\int_{0}^{s}\tilde{\ell}(s,u)\mathrm{d}\tilde{W}_{u}+g(s)-\int_{0}^{s}\tilde{\ell}(s,u)g(u)\,\mathrm{d}u\right]\mathrm{d}\tilde{W}_{s}$
$\displaystyle-\frac{1}{2}\int_{0}^{t}\left[\int_{0}^{s}\tilde{\ell}(s,u)\mathrm{d}\tilde{W}_{u}+g(s)-\int_{0}^{s}\tilde{\ell}(s,u)g(u)\,\mathrm{d}u\right]^{2}\mathrm{d}s\bigg{\\}}$

Let then $\tilde{X}$ be a Gaussian process. Set

(15) $\tilde{W}_{t}=\int_{0}^{t}L^{-1}(t,s)\,\mathrm{d}\tilde{X}_{s}.$

By the transfer principle Theorem 2 and Hitsuda representation theorem we then
have the following Cameron–Martin–Girsanov–Hitsuda theorem.

######  Proposition 3.

A Gaussian process $\tilde{X}$ is equivalent in law to a ccmfBm if and only if
the process $\tilde{W}$ given by (15) is of from (10) and in this case the
process

(16) $X_{t}=\int_{0}^{t}L(t,s)\,\mathrm{d}W_{s}$

is a ccmfBm, where $W$ is a Bm constructed from $\tilde{X}$ via (15) and (11).
The likelihood ratio $\varphi_{t}$ between $\tilde{X}$ and $X$ given the
observations on $[0,t]$ is given by (4) and (4).

The Hitsuda representation can also be given without constructing the
intermediate Bm’s as follows:

######  Proposition 4.

A Gaussian process $\tilde{X}$ is equivalent in law to the ccmfBm $X$ if and
only if it admits the representation

$\tilde{X}_{t}=X_{t}-\int_{0}^{t}f(t,s)\,\mathrm{d}X_{s}-\mathrm{L}g(t),$

where $g\in L^{2}([0,T])$ and the Voterra kernel $f$ is defined by the
equation

(17) $\mathrm{L}^{*}f(t,\cdot)(s)=\mathrm{L}\ell(\cdot,s)(t)$

for some Volterra kernel $\ell\in L^{2}([0,T]^{2})$.

######  Remark 4.

If $X$ is the Bm, then $\mathrm{L}=\mathrm{I}$ is simply the integral operator
$\mathrm{I}f(t)=\int_{0}^{t}f(s)\mathrm{d}s$ and its adjoint
$\mathrm{L}^{*}=\mathrm{I}^{*}$ is simply the identity operator. Thus for the
Bm the equation (17) takes the familiar form of the Hitsuda’s representation
theorem

$f(t,s)=\int_{s}^{t}\ell(u,s)\,\mathrm{d}u.$

for some $\ell\in L^{2}([0,T]^{2})$.

The Cameron–Martin–Girsanov likelihood $\varphi_{t}$ can also be given without
constructing the intermediate Bm’s by using Skorokhod integrals. We omit this
and instead consider a simple drift estimation case as an application of the
Cameron–Martin–Girsanov theorem.

Suppose we want to test the following hypotheses given observations on the
time interval $[0,t]$:

  * $H_{0}$

The observation comes from a centered ccmfBm $X_{s}$, $s\in[0,t]$, with known
parameters $a,b,H$.

  * $H_{\theta}$

The observation comes from a drifting ccmfBm $X_{s}+\theta s$, $s\in[0,t]$,
with known parameters $a,b,H$ but unknown drift $\theta$.

Then the likelihood ratio between these hypothesis is

$\displaystyle\varphi_{t}(\theta)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{d}\mathbb{P}_{\theta}}{\mathrm{d}\mathbb{P}_{0}}\Big{|}\mathscr{F}_{t}$
$\displaystyle=$ $\displaystyle\exp\left\\{\theta
w_{t}-\frac{\theta^{2}}{2}t\right\\},$

where $w$ is constructed from the observations $x$ as

$w_{t}=\int_{0}^{t}L^{-1}(t,s)\,\mathrm{d}x_{s}$

This leads to the maximum likelihood estimator of the drift $\theta$ given
observations $x_{s}$, $s\in[0,t]$:

$\hat{\theta}_{t}=\frac{1}{t}\int_{0}^{t}L^{-1}(t,s)\,\mathrm{d}x_{s}.$

##  5\. Prediction

Let $\mathscr{F}_{u}^{X}=\sigma\\{X_{u};u\leq t\\}$ denote the
$\sigma$-algebra of observing the ccmfBm over the interval $[0,t]$. Naturally,
we are interested in predicting the future, i.e., we are interested in the
conditional future probability law of the process $X$ given the information
$\mathscr{F}_{u}^{X}$. The transfer principle of Theorem 2 provides us these
prediction formulas for the ccmfBm in the same way as in [21, 22]:

######  Theorem 3.

The conditional process $t\mapsto X_{t}(u)=X_{t}|\mathscr{F}_{u}^{X}$, $t\geq
u$, is Gaussian with stochastic mean

$\displaystyle\hat{m}_{t}(u)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[X_{t}\big{|}\mathscr{F}_{u}^{X}\right]$
$\displaystyle=$ $\displaystyle
X_{u}+\int_{0}^{u}\Psi(t,s|u)\,\mathrm{d}X_{s},$

where

$\Psi(t,s|u)=(\mathrm{L}^{*})^{-1}\left[L(t,\cdot)-L(u,\cdot)\right](s);$

and with deterministic covariance

$\displaystyle\hat{R}(t,s|u)$ $\displaystyle=$
$\displaystyle\mathbb{C}\mathrm{ov}\left[X_{t},X_{s}\big{|}\mathscr{F}_{u}^{X}\right]$
$\displaystyle=$ $\displaystyle R(s,t)-\int_{0}^{u}L(t,v)L(s,v)\,\mathrm{d}v.$

##  6\. Simulation

We illustrate the long-range dependent ccmfBm by simulating its paths. The
paths are simulated on $N=500$ equidistant time points $t_{k}=k/N$ on the
interval $[0,1]$ by using the Cholesky decomposition
$\mathbf{R}_{H}=\mathbf{L}_{H}\mathbf{L}_{H}^{\top}$ of the covariance matrix
$\mathbf{R}_{H}(k,j)=R_{H}(t_{k},t_{j})$ of the fBm:

(18)
$X_{t_{k}}=\frac{a}{\sqrt{N}}\sum_{j=1}^{k}\xi_{j}+b\sum_{j=1}^{k}\mathbf{L}_{H}(k,j)\xi_{j},$

where $\xi_{j}$’s are i.i.d. standard random variables.

######  Remark 5.

Equation (18) provides exact simulation. The cost of having exact simulation
is of course in calculating the Cholesky decomposition. If one is happy to
make some error in the simulation, one can use the Molchan–Golosov
representation of the fBm for simulation. One way of doing this is to use the
approximation

(19)
$X_{t}\approx\sum_{j=1}^{[Nt]}\left[a+bN\int_{\frac{j-1}{N}}^{\frac{j}{N}}K_{H}\left(\frac{[Nt]}{N},s\right)\,\mathrm{d}s\,\right]\frac{1}{\sqrt{N}}\xi_{j}$

and approximate the integral above in some way. If the integral is
approximated in an efficient way, then this approach can be fast as it avoids
calculating the Cholesky decomposition and instead uses the Molchan–Golosov
kernel $K_{H}$ as a proxy for the Cholesky square root. See [17] for more
information on the convergence of the approximation (19).

######  Remark 6.

The integral representation

$X_{t}=\int_{0}^{t}L(t,s)\,\mathrm{d}W_{s}$

provides us with a series expansion. Indeed, let
$(\tilde{e}_{k})_{k=1}^{\infty}$ be your favorite orthonormal basis on
$L^{2}([0,T])$ (note that the basis functions $\tilde{e}_{k}$ depend on $T$)
and let $(\xi_{k})_{k=1}^{\infty}$ be a sequence of i.i.d. standard normal
random variables. Let

(20) $e_{k}(t)=\int_{0}^{t}L(t,s)\tilde{e}_{k}(s)\,\mathrm{d}s.$

Then it follows that

(21) $X_{t}=\sum_{k=1}^{\infty}e_{k}(t)\xi_{k},$

where the series (20) converges both in $L^{2}$ and pointwise. See [8] for
details and for an explicit series expansion for the fBm that be extended to
the ccmfBm in a straightforward manner. Cutting the series (21) and
approximating the integral (20) provides us yet another way to approximately
simulate the ccmfBm process.

In figures 4–6 we have plotted simulated paths of the completely correlated
mixed fractional Brownian motion (ccmfBm) together with its components, the
Brownian motion (Bm) and the completely correlated fractional Brownian motion
(ccfBm) that is constructed from the Bm. The plots were created by using the
exact simulation formula (18).

Figure 4. ccmfBm with $a=0.4$, $b=1.4$, $H=0.6$. Figure 5. ccmfBm with $a=1$,
$b=3$, $H=0.75$. Figure 6. ccmfBm with $a=4$, $b=9$, $H=0.9$.

Figures 4–6 illustrate that the Hölder continuity of the ccmfBm is the same as
the Hölder continuity of the driving Bm.

##  7\. Conclusions and Discussion

We have considered the long-range dependent completely correlated mixed
fractional Brownian motion (ccmfBm) $X$ that is constructed by using a single
Brownian motion (Bm) $W$ and a complete correlated fractional Brownian motion
(ccfBm) $B^{H}$ as

(22) $\displaystyle X_{t}$ $\displaystyle=$ $\displaystyle aW_{t}+bB^{H}_{t}$
$\displaystyle=$ $\displaystyle
aW_{t}+b\int_{0}^{t}K_{H}(t,s)\,\mathrm{d}W_{s},$

where $K_{H}(t,s)$ is the so-called Molchan–Golosov kernel.

We have shown that the short-time path behavior (Hölder continuity and
quadratic variation) of the ccmfBm are the same as those of the Bm, but unlike
the Bm, the ccmfBm has long-range dependence that is characterized by the fBm
part. We have constructed explicitly the transfer principle for the ccmfBm,
i.e., we have constructed explicitly a kernel $L^{-1}$ such that given the
ccmfBm $X$ the driving Bm $W$ in (22) can be recovered from the equation

$W_{t}=\int_{0}^{t}L^{-1}(t,s)\,\mathrm{d}X_{s}.$

We also noted that the transfer principle for ccmfBm has a more convenient
form than the corresponding principle of independent mixture mfBm. Indeed, our
inverse kernel $L^{-1}$ have a series expansion that converges fast, while for
the independent mfBm it is only known that such an inverse kernel exists and
it is a solution of a certain integral equation, see [5]. We have considered
some applications of the transfer principle like parameter estimation and
prediction. We have illustrated the ccmfBm by simulations.

Finally, let us discuss shortly the short-range dependent ccmfBm. In the
short-range dependent case when $H\in(0,1/2)$ the Molchan–Golosov kernel
$K_{H}$ takes a different form and the adjoint associated operator
$\mathrm{L}^{*}$ of the kernel $L=a\mathbf{1}+bK_{H}$ is no longer a bounded
operator on $L^{2}$. Consequently, Lemma 1 is no longer true. In particular,
the integrand space $\Lambda$ will be a strict subset of $L^{2}$ and the
transfer principle of Theorem 2 is no longer true without radical
modifications. Also, the short time-scale behavior of the short-range
dependent ccmfBm will be governed by the fBm part, not the Brownian part: $H$
will be the Hölder index of the ccmfBm and the quadratic variation will be
infinite.

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# Some matrix inequalities of log-majorization type

Bo-Yan Xi  College of Mathematics, Inner Mongolia University for
Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Fuzhen Zhang  Department of
Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale,
FL 33314, USA<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is two-fold: we present some matrix inequalities of
log-majorization type for eigenvalues indexed by a sequence; we then apply our
main theorem to generalize and improve the Hua-Marcus’ inequalities. Our
results are stronger and more general than the existing ones.

###### Key words and phrases:

Eigenvalue; Hua’s determinant inequality; majorization inequality

###### 2020 Mathematics Subject Classification:

15A42, 15A45, 47A63

Xi’s work was partially supported by NNSF of China grant No. 11361038.

##  1\. Introduction

Let $\mathbb{C}^{m\times n}$ be the space of $m\times n$ complex matrices. For
$A\in\mathbb{C}^{n\times n}$, we denote the eigenvalues of $A$ by
$\lambda_{1}(A),\lambda_{2}(A),\dots,\lambda_{n}(A)$ and the singular values
of $A$ by $\sigma_{1}(A),\sigma_{2}(A),\dots,\sigma_{n}(A)$. If
$\lambda_{1}(A),\lambda_{2}(A),\dots,\lambda_{n}(A)$ are all real, we arrange
them in decreasing order:
$\lambda_{1}(A)\geq\lambda_{2}(A)\geq\cdots\geq\lambda_{n}(A)$. The singular
values are always ordered decreasingly:
$\sigma_{1}(A)\geq\sigma_{2}(A)\geq\dots\geq\sigma_{n}(A)$.

Horn conjecture …

There is a large family of matrix inequalities concerning the eigenvalues and
singular values of the product (including Schur or Hadamard product) and sum
of matrices. These inequalities may be sorted in four types:
$\sum\zeta_{{}_{\mathcal{I}}}(A+B),\,\prod\zeta_{{}_{\mathcal{I}}}(A+B),\,\sum\zeta_{{}_{\mathcal{I}}}(AB),\,\prod\zeta_{{}_{\mathcal{I}}}(AB)$,
where $A$ and $B$ are generic matrices, and $\zeta_{{}_{\mathcal{I}}}(\cdot)$
represents selected eigenvalues or singular values indexed by a sequence
${\mathcal{I}}$. The inequalities in sum $\sum$ are usually called
majorization type ([7, p. 45], [13]), while the ones in product $\prod$ are
referred to as logarithmic (log-) majorization type. See [13, p. 16], [3, 9]).
For example, the following inequalities [6] (or [13, G.2.a, p. 333]) are log-
majorization type with $\mathcal{I}=\\{k,\dots,n\\}$: for each $k=1,\dots,n$,

$\prod\limits_{t=k}^{n}\lambda_{t}(A+B)\geq\prod\limits_{t=k}^{n}[\lambda_{t}(A)+\lambda_{t}(B)]\geq\prod\limits_{t=k}^{n}\lambda_{t}(A)+\prod\limits_{t=k}^{n}\lambda_{t}(B),$
(1.1)

where $A$ and $B$ are $n\times n$ positive semidefinite matrices.

Closely related inequalities of the same type, due to Oppenheim [14] (or [13,
F.2, p. 685]), are, for $n\times n$ positive semidefinite $A$ and $B$, and for
each $k=1,\dots,n,$

$\displaystyle\left[\prod\limits_{t=k}^{n}\lambda_{t}(A+B)\right]^{\frac{1}{n-k+1}}\geq$
$\displaystyle\left[\prod\limits_{t=k}^{n}\lambda_{t}(A)\right]^{\frac{1}{n-k+1}}+\left[\prod\limits_{t=k}^{n}\lambda_{t}(B)\right]^{\frac{1}{n-k+1}},$
(1.2)

which give the Minkowski inequality by setting $k=1$ ([13, p. 685] or [23, p.
215])

$\displaystyle\left[\det(A+B)\right]^{\frac{1}{n}}\geq\left[\det(A)\right]^{\frac{1}{n}}+\left[\det(B)\right]^{\frac{1}{n}}.$
(1.3)

In [20], we showed majorization inequalities of
$\sum\zeta_{{}_{\mathcal{I}}}(AB)$ for Hermitian matrices $A$ and $B$ with an
arbitrary index set $\mathcal{I}$. In this paper, we present some log-
majorization inequalities of $\prod\zeta_{{}_{\mathcal{I}}}(A+B)$ with any
index set $\mathcal{I}$ for positive semidefinite matrices $A$ and $B$,
extending (1.1) and (1.2). Our theorems are more general and stronger than
some existing results. We apply main results to improve the Hua-Marcus
inequalities for contractive matrices.

##  2\. Some lemmas

Let $X^{*}$ denote the conjugate transpose of matrix or vector $X$. For a
square matrix $A$, we write $A\geq 0$ if $A$ is positive semidefinite and
$A>0$ if $A$ is positive definite. For Hermitian matrices
$A,B\in\mathbb{C}^{n\times n}$, we write $A\geq B$ if $A-B\geq 0$.

Through the paper, let $n$ be a positive integer. Let
$\mathcal{I}=\\{i_{1},\dots,i_{k}\\}$, where $1\leq i_{1}<\cdots<i_{k}\leq n$
is any subsequence of $1,\dots,n$, $k=1,\dots,n$. If $A\in\mathbb{C}^{n\times
n}$ is positive semidefinite, then
$\lambda_{i_{1}}(A)\geq\cdots\geq\lambda_{i_{k}}(A)$ are $k$ eigenvalues of
$A$ indexed by $\mathcal{I}=\\{i_{1},\dots,i_{k}\\}$. For $t=1,\dots,k$,
setting $i_{t}=t$ gives the first $k$ largest eigenvalues of $A$; putting
$i_{t}=n-k+t$ gives the last $k$ smallest eigenvalues of $A$.

######  Lemma 2.1 (Hoffman [1, Cor. 2.5]).

Let $A\in\mathbb{C}^{n\times n}$ be positive semidefinite. Then

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A)$ $\displaystyle=$
$\displaystyle\max\limits_{\mathbb{S}_{1}\subset\cdots\subset\mathbb{S}_{k}\subset\mathbb{C}^{n}\hfill\atop\dim\mathbb{S}_{t}=i_{t}}\min\limits_{{x}_{t}\in\mathbb{S}_{t},({x}_{r},{x}_{s})=\delta_{rs}\hfill\atop
U_{k}=\left(x_{1},\dots,x_{k}\right)}\det({U_{k}^{*}AU_{k}}),$

where $\delta_{rs}$ is the Kronecker delta, i.e., $\delta_{rs}=1$ if $r=s$, or
0 otherwise.

######  Lemma 2.2 (Lidskiǐ [11]).

Let $A,B\in\mathbb{C}^{n\times n}$ be positive semidefinite. Then

$\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A)\lambda_{n-t+1}(B)\leq\prod\limits_{t=1}^{k}\lambda_{i_{t}}(AB)\leq\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A)\lambda_{t}(B).$
(2.1)

More general inequalities of (2.1) for singular values are due to Gel’fand and
Naimark (see, e.g., [13, p. 340]). The inequalities on the left-hand side of
(2.1) can be found explicitly in [17].

######  Lemma 2.3.

Let $a_{1},\dots,a_{n}$ and $b_{1},\dots,b_{n}$ be nonnegative real numbers.
Then

$\prod\limits_{t=1}^{n}(a_{t}+b_{t})\geq\prod\limits_{t=1}^{n}a_{t}+\prod\limits_{t=1}^{n}b_{t}+(2^{n}-2)\left[\prod\limits_{t=1}^{n}(a_{t}b_{t})\right]^{\frac{1}{2}}.$
(2.2)

In particular, for $x,y\geq 0$,

$\displaystyle\big{(}x^{\frac{1}{n}}+y^{\frac{1}{n}}\big{)}^{n}\geq
x+y+\left(2^{n}-2\right)(xy)^{\frac{1}{2}}.$ (2.3)

###### Proof.

Use induction on $n$ and apply the arithmetic-geometric mean inequality. ∎

We remark that Lemma 2.3 implies immediately a result of Hartfiel [8]:

$\det(A+B)\geq\det A+\det B+(2^{n}-2)[\det A\det B]^{\frac{1}{2}}$ (2.4)

because of the fact that any two positive semidefinite matrices of the same
size are simultaneously ∗-congruent to diagonal matrices (see, e.g., [23, p.
209]).

##  3\. Main Results

We begin with a result that is important to the proof of our main theorem.

######  Lemma 3.1.

Let $D={\rm{diag}}\big{(}\lambda_{1},\lambda_{2},\dots,\lambda_{n}\big{)}$,
where $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$, and let
$U_{k}=\left(u_{1},\dots,u_{k}\right)=\left(u_{ij}\right)$ be a partial
isometry, i.e., $U_{k}$ is an $n\times k$ matrix such that
$U^{*}_{k}U_{k}=I_{k}$. If $m\geq 1$ and $u_{i1}=0$ for all $i>m$, then

$\displaystyle\det(U^{*}_{k}DU_{k})\geq\lambda_{m}\det\left[\left(u_{2},\dots,u_{k}\right)^{*}D_{m}\left(u_{2},\dots,u_{k}\right)\right],$
(3.1)

where
$D_{m}={\rm{diag}}\big{(}\underbrace{\lambda_{m},\dots,\lambda_{m}}_{m},\lambda_{m+1},\dots,\lambda_{n}\big{)}.$

###### Proof.

Let $V=\left(u_{2},\dots,u_{k}\right)$. Since $D\geq D_{m}$ and $u_{i1}=0$ for
$i>m$, we have

$\displaystyle\det(U_{k}^{*}DU_{k})$ $\displaystyle\geq$
$\displaystyle\det(U_{k}^{*}D_{m}U_{k})=\det\begin{pmatrix}\lambda_{m}u^{*}_{1}u_{1}&\lambda_{m}u^{*}_{1}V\\\
\lambda_{m}V^{*}u_{1}&V^{*}D_{m}V\end{pmatrix}$ $\displaystyle=$
$\displaystyle\det\begin{pmatrix}\lambda_{m}&0\\\
0&V^{*}D_{m}V\end{pmatrix}=\lambda_{m}\det\left(V^{*}D_{m}V\right)$
$\displaystyle=$
$\displaystyle\lambda_{m}\det\left[\left(u_{2},\dots,u_{k}\right)^{*}D_{m}\left(u_{2},\dots,u_{k}\right)\right].$

∎

We are ready to present our main result.

######  Theorem 3.1.

Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. Then

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}^{\frac{1}{k}}(A+B)\geq\prod\limits_{t=1}^{k}\lambda_{i_{t}}^{\frac{1}{k}}(A)+\prod\limits_{t=1}^{k}\lambda_{n-t+1}^{\frac{1}{k}}(B)$
(3.2)

and

$\displaystyle{\small\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A+B)\geq\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A)+\prod\limits_{t=1}^{k}\lambda_{n-t+1}(B)+(2^{k}-2)\prod\limits_{t=1}^{k}\left[\lambda_{i_{t}}(A)\lambda_{n-t+1}(B)\right]^{\frac{1}{2}}\\!.\;}$
(3.3)

###### Proof.

If $n=1$, the inequalities become equalities and hold trivially. Let $n\geq
2$. By spectral decomposition, there exists a unitary matrix
$U=(u_{1},u_{2},\dots,u_{n})\in\mathbb{C}^{n\times n}$, where
$u_{1},u_{2},\dots,u_{n}$ are orthonormal eigenvectors associated with
$\lambda_{1}(A),\lambda_{2}(A),\dots,$ $\lambda_{n}(A)$, respectively, such
that

$A=U\,{\rm{diag}}(\lambda_{1}(A),\lambda_{2}(A),\dots,\lambda_{n}(A))U^{*}.$

For each $i_{t}$ in the sequence $i_{1}<i_{2}<\cdots<i_{k}$, let
$\mathbb{S}^{(0)}_{t}={\rm{Span}}(u_{1},u_{2},\ldots,u_{i_{t}}),$
$t=1,2,\dots,k.$ Then
$\mathbb{S}^{(0)}_{1}\subset\mathbb{S}^{(0)}_{2}\subset\cdots\subset\mathbb{S}^{(0)}_{k}$
and $\dim\big{(}\mathbb{S}^{(0)}_{t}\big{)}={i_{t}},$ $t=1,2,\dots,k.$

Let $\\{x_{1},\dots,x_{k}\\}$ be any set of orthonormal vectors, where
$x_{t}\in\mathbb{S}^{(0)}_{t},$ $t=1,2,\dots,k.$ Let
$x_{t}=a_{1t}u_{1}+\cdots+a_{i_{t}t}u_{i_{t}}+0u_{i_{t}+1}+\dots+0u_{i_{k}}=(u_{1},\dots,u_{i_{k}})\alpha_{t}$,
where $\alpha_{t}=(a_{1t},\dots,a_{i_{t}t},0,\dots,0)^{\tiny T}$ (here T is
for transpose), $t=1,2,\dots,k$. Then

$U_{k}=(x_{1},\dots,x_{k})=(u_{1},\dots,u_{i_{k}})(\alpha_{1},\dots,\alpha_{k}).$

Let $V_{k}=(\alpha_{1},\dots,\alpha_{k})\in\mathbb{C}^{i_{k}\times k}$. Since
$\\{x_{1},\dots,x_{k}\\}$ and $\\{u_{1},\dots,u_{i_{k}}\\}$ are orthonormal
sets, we see that $V_{k}^{*}V_{k}=I_{k}$. So $V_{k}$ is a partial isometry,
and the components of the first column of $V_{k}$ are zeros except the first
$i_{1}$ components (i.e., the last $i_{k}-i_{1}$ components of $\alpha_{1}$
are all equal to zero).

In Lemma 3.1, setting $n=i_{k}$, $m=i_{1}$, $U_{k}=V_{k}$, and applying (3.1)
to the $i_{k}\times i_{k}$ matrix $D_{i_{1}}=\mathop{\rm
diag}(\underbrace{\lambda_{i_{1}}(A),\dots,\lambda_{i_{1}}(A)}_{i_{1}}\,,\lambda_{{i_{1}}+1}(A),\dots,\lambda_{i_{k}}(A)),$
we obtain

$\displaystyle\det(U^{*}_{k}AU_{k})$ $\displaystyle=$
$\displaystyle\det\big{[}(\alpha_{1},\dots,\alpha_{k})^{*}(u_{1},\dots,u_{i_{k}})^{*}A(u_{1},\dots,u_{i_{k}})(\alpha_{1},\dots,\alpha_{k})\big{]}$
$\displaystyle=$
$\displaystyle\det\big{[}(\alpha_{1},\dots,\alpha_{k})^{*}\mathop{\rm
diag}(\lambda_{1}(A),\dots,\lambda_{i_{k}}(A))(\alpha_{1},\dots,\alpha_{k})\big{]}$
$\displaystyle=$ $\displaystyle\det\big{[}V_{k}^{*}\mathop{\rm
diag}(\lambda_{1}(A),\dots,\lambda_{i_{k}}(A))V_{k}\big{]}$
$\displaystyle\geq$
$\displaystyle\lambda_{i_{1}}(A)\det\big{[}(\alpha_{2},\dots,\alpha_{k})^{*}D_{i_{1}}(\alpha_{2},\dots,\alpha_{k})\big{]}.$

Repeatedly using Lemma 3.1, we get

$\det(U_{k}^{*}AU_{k})\geq\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A).$ (3.4)

A minimax eigenvalue result of Fan (see [4, 5] or [13, A.3.a, p. 787]) ensures

$\displaystyle\det(U_{k}^{*}BU_{k})\geq\min_{U^{*}U=I_{k}}\det(U^{*}BU)=\prod\limits_{t=1}^{k}\lambda_{n-t+1}(B).$

An application of the Minkowski inequality (1.3) together with (3.4) reveals

$\displaystyle\det\big{[}{U_{k}^{*}(A+B)U_{k}}\big{]}$ $\displaystyle=$
$\displaystyle\det({U_{k}^{*}AU_{k}+U_{k}^{*}BU_{k}})$ $\displaystyle\geq$
$\displaystyle\left[\big{(}\det({U_{k}^{*}AU_{k}})\big{)}^{\frac{1}{k}}+\big{(}\det({U_{k}^{*}BU_{k}})\big{)}^{\frac{1}{k}}\right]^{k}$
$\displaystyle\geq$
$\displaystyle\left[\prod\limits_{t=1}^{k}\lambda^{\frac{1}{k}}_{i_{t}}(A)+\prod\limits_{t=1}^{k}\lambda^{\frac{1}{k}}_{n-t+1}(B)\right]^{k}.$

Thus, by Lemma 2.1, we derive (3.2) and (3.3) as follows:

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A+B)$ $\displaystyle=$
$\displaystyle\max\limits_{\mathbb{S}_{1}\subset\cdots\subset\mathbb{S}_{k}\subset\mathbb{C}^{n}\hfill\atop\dim\mathbb{S}_{t}=i_{t}}\min\limits_{{y}_{t}\in\mathbb{S}_{t},\,({y}_{r},{y}_{s})=\delta_{rs}\hfill\atop
W_{k}=\left(y_{1},\dots,y_{k}\right)}\det\big{[}{W_{k}^{*}(A+B)W_{k}}\big{]}$
$\displaystyle\geq$
$\displaystyle\min\limits_{{x}_{t}\in\mathbb{S}^{(0)}_{t}\,({x}_{r},{x}_{s})=\delta_{rs}\hfill\atop
U_{k}=\left(x_{1},\dots,x_{k}\right)}\det\big{[}{U_{k}^{*}(A+B)U_{k}}\big{]}$
$\displaystyle\geq$
$\displaystyle\left[\prod\limits_{t=1}^{k}\lambda^{\frac{1}{k}}_{i_{t}}(A)+\prod\limits_{t=1}^{k}\lambda^{\frac{1}{k}}_{n-t+1}(B)\right]^{k}$
$\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A)+\prod\limits_{t=1}^{k}\lambda_{n-t+1}(B)$
$\displaystyle+(2^{k}-2)\prod\limits_{t=1}^{k}\left[\lambda_{i_{t}}(A)\lambda_{n-t+1}(B)\right]^{\frac{1}{2}}.$

The last inequality is by (2.3) with
$x=\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A)$ and
$y=\prod\limits_{t=1}^{k}\lambda_{n-t+1}(B)$. ∎

In (3.2), letting $i_{t}=n-k+t$, $t=1,2,\ldots,k$, we arrive at the
inequalities (1.2) of Oppenhiem for the product of $k$ smallest eigenvalues.
Setting $i_{t}=t$, $t=1,2,\ldots,k$, we obtain analogous inequalities of 1.2
for the product of $k$ largest eigenvalues.

######  Corollary 3.1.

Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. Then

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{t}^{\frac{1}{k}}(A+B)\geq\prod\limits_{t=1}^{k}\lambda_{t}^{\frac{1}{k}}(A)+\prod\limits_{t=1}^{k}\lambda_{n-t+1}^{\frac{1}{k}}(B).$
(3.5)

What follows is a lower bound for the product of any two eigenvalues of the
sum in terms of the eigenvalues of individual matrices.

######  Corollary 3.2.

Let $A$ and $B$ be $n\times n$ positive semidefinite matrices. Then

$\displaystyle\lambda_{i}(A+B)\lambda_{j}(A+B)$ $\displaystyle\geq$
$\displaystyle\lambda_{i}(A)\lambda_{j}(A)+\lambda_{n-1}(B)\lambda_{n}(B)$
$\displaystyle+2\big{[}\lambda_{i}(A)\lambda_{j}(A)\lambda_{n-1}(B)\lambda_{n}(B)\big{]}^{\frac{1}{2}}.$

Remark: In view of Fiedler’s (1.1) and inequalities (3.3), it is tempting to
have

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}(A+B)\geq\prod\limits_{t=1}^{k}\big{[}\lambda_{i_{t}}(A)+\lambda_{n-t+1}(B)\big{]}.$
(3.6)

However, this need not be true. Take $A=B=\left(\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}\right)$, $k=2,i_{1}=1,i_{2}=2$. Then the left-hand side
of (3.6) is 0, while the right-hand side is 1. (Note: it is always true that
$\lambda_{i}(A+B)\geq\lambda_{i}(A)+\lambda_{n}(B)$. See, e.g., [23, p. 274]).

The following inequality in (3.7) is Fiedler’s 1st inequality in (1.1).
Inequality in (3.3) is stronger than Fiedler’s 2nd inequality. (3.3) is proved
by Lemma 2.3. The inequalities in (3.3) are immediate from the inequalities
(3.3) in the theorem.

######  Corollary 3.3.

Let $A,B\in\mathbb{C}^{n\times n}$ be positive semidefinite matrices. Then

$\displaystyle\prod\limits_{t=n-k+1}^{n}\lambda_{t}(A+B)$ $\displaystyle\geq$
$\displaystyle\prod\limits_{t=n-k+1}^{n}\big{[}\lambda_{t}(A)+\lambda_{t}(B)\big{]}$
(3.7) $\displaystyle\geq$
$\displaystyle\prod\limits_{t=n-k+1}^{n}\lambda_{t}(A)+\prod\limits_{t=n-k+1}^{n}\lambda_{t}(B)$
$\displaystyle+\left(2^{k}-2\right)\prod\limits_{t=n-k+1}^{n}\left[\lambda_{t}(A)\lambda_{t}(B)\right]^{\frac{1}{2}}$

and

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{t}(A+B)$ $\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{t}(A)+\prod\limits_{t=1}^{k}\lambda_{n-t+1}(B)$
$\displaystyle+\left(2^{k}-2\right)\prod\limits_{t=1}^{k}\left[\lambda_{t}(A)\lambda_{n-t+1}(B)\right]^{\frac{1}{2}}.$

Remark: In view of (3.7), it is appealing in the display (3.3 to have

$\prod\limits_{t=1}^{k}\lambda_{t}(A+B)\;\;\geq\;\;\prod\limits_{t=1}^{k}\big{[}\lambda_{t}(A)+\lambda_{t}(B)\big{]}\;\;\mbox{or}\;\;\prod\limits_{t=1}^{k}\big{[}\lambda_{t}(A)+\lambda_{n-t+1}(B)\big{]}.$

But neither one is true. Let $A=\left(\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}\right)$, $B=\left(\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}\right)$, $k=1$. Then
$\lambda_{1}(A+B)=1<2=\lambda_{1}(A)+\lambda_{1}(B).$ See the counterexample
below (3.6) for the second case.

##  4\. Hua-Marcus inequalities for contractions

A matrix $A\in\mathbb{C}^{m\times n}$ is said to be (strictly) _contractive_
if $I_{n}-A^{*}A\geq 0$ ($>0$), equivalently, the largest singular value
(i.e., the spectral norm) of $A$ is less than or equal to (resp. $<$) $1$. We
use $\mathcal{C}_{m\times n}$ to denote the set of $m\times n$ contractive
matrices and $\mathcal{SC}_{m\times n}$ to denote the set of $m\times n$
strictly contractive matrices.

In this section we apply our main theorem to derive some inequalities of Hua-
Marcus type for contractive matrices. We begin by citing Hua’s results in
[10].

######  Theorem 4.1 (Hua [10, Theorems 1 and 2]).

Let $A,B\in\mathcal{SC}_{n\times n}$. Then

$\displaystyle\det(I-A^{*}A)\det(I-B^{*}B)+\left|\det(A-B)\right|^{2}\leq\left|\det(I-A^{*}B)\right|^{2}.$
(4.1)

Consequently,

$\displaystyle\det(I-A^{*}A)\det(I-B^{*}B)\leq\left|\det(I-A^{*}B)\right|^{2}.$
(4.2)

Equality in (4.1) holds if and only if $A=B$.

The following [23, Theorem 7.18] is a reversal of the Hua inequality 4.1 for
general $n\times n$ matrices $A$ and $B$ that need not be contractive:

$\displaystyle\left|\det(I-A^{*}B)\right|^{2}\leq\det(I+A^{*}A)\det(I+B^{*}B)-\left|\det(A+B)\right|^{2}.$
(4.3)

Marcus [12] extended the Hua inequality (4.2) to the inequalities of
eigenvalues.

######  Theorem 4.2 (Marcus [12, Theorem]).

Let $A,B\in\mathcal{C}_{n\times n}$. Then

$\displaystyle\prod\limits_{t=1}^{k}|\lambda_{n-t+1}(I-A^{*}B)|^{2}\geq\prod\limits_{t=1}^{k}\big{[}1-\lambda_{t}(A^{*}A)\big{]}\big{[}1-\lambda_{t}(B^{*}B)\big{]}.$
(4.4)

Since
$|\det(I-A^{*}B)|=\prod\limits_{t=1}^{n}|\lambda_{n-t+1}(I-A^{*}B)|=\prod\limits_{t=1}^{n}\sigma_{n-t+1}(I-A^{*}B)$,
by Weyl’s log-majorization inequality ([18] or [13, p. 317]), the eigenvalues
in absolute values are log-majorized by the singular values, we get the
stronger inequality than (4.4):

$\displaystyle\prod\limits_{t=1}^{k}\sigma^{2}_{n-t+1}(I-A^{*}B)\geq\prod\limits_{t=1}^{k}\big{[}1-\lambda_{t}(A^{*}A)\big{]}\big{[}1-\lambda_{t}(B^{*}B)\big{]}.$
(4.5)

Related inequalities of Hua-Marcus type are seen in [2, 14, 15, 16, 21, 22].

The following result is stronger and more general than 4.5.

######  Theorem 4.3.

Let $A,B\in\mathcal{SC}_{n\times n}$. Then

$\displaystyle\prod\limits_{t=1}^{k}\sigma^{2}_{i_{t}}\left(I-A^{*}B\right)$
$\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\big{[}1-\lambda_{t}(A^{*}A)\big{]}\big{[}1-\lambda_{n-i_{t}+1}(B^{*}B)\big{]}$
(4.7)
$\displaystyle+\prod\limits_{t=1}^{k}\frac{\big{[}1-\lambda_{t}(A^{*}A)\big{]}\,\sigma^{2}_{n-t+1}(A-B)}{1-\lambda_{n-t+1}(A^{*}A)}$
$\displaystyle+(2^{k}-2)\prod\limits_{t=1}^{k}\big{[}1-\lambda_{t}(A^{*}A)\big{]}\big{[}\lambda_{i_{t}}(F)\lambda_{n-t+1}(H)\big{]}^{\frac{1}{2}}\\!,\;$

where $F=I-B^{*}B$ and $H=(A-B)^{*}(I-AA^{*})^{-1}(A-B).$

###### Proof.

For $A,B\in\mathcal{SC}_{n\times n}$, let

$F=I-B^{*}B\;\;\;\mbox{and}\;\;\;H=(A-B)^{*}(I-AA^{*})^{-1}(A-B).$

Then (see [10, Theorem 1] or [23, pp. 230-231])

$F+H=(I-B^{*}A)(I-A^{*}A)^{-1}(I-A^{*}B).$ (4.8)

Applying the left inequality of (2.1) with $i_{t}=n-t+1$ to $H$ reveals

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{n-t+1}(H)$ $\displaystyle=$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{n-t+1}\big{[}(I-AA^{*})^{-1}(A-B)(A-B)^{*}\big{]}$
(4.9) $\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{n-t+1}\big{[}(I-AA^{*})^{-1}\big{]}\sigma^{2}_{n-t+1}\big{(}A-B\big{)}$
$\displaystyle=$
$\displaystyle\prod\limits_{t=1}^{k}[1-\lambda_{n-t+1}(A^{*}A)]^{-1}\sigma^{2}_{n-t+1}\big{(}A-B\big{)}.$

Applying the right inequality of (2.1) to $F+H$ in the product form in (4.8)
yields

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}\left(F+H\right)\leq\prod\limits_{t=1}^{k}\left[1-\lambda_{t}(A^{*}A)\right]^{-1}\sigma^{2}_{i_{t}}\left(I-A^{*}B\right).$
(4.10)

Thus

$\displaystyle\prod\limits_{t=1}^{k}\sigma^{2}_{i_{t}}\left(I-A^{*}B\right)\geq\prod\limits_{t=1}^{k}\left[1-\lambda_{t}(A^{*}A)\right]\lambda_{i_{t}}\left(F+H\right).$
(4.11)

By inequalities (3.3) and (4.9), we obtain

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}\left(F+H\right)$ (4.12)
$\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}(F)+\prod\limits_{t=1}^{k}\lambda_{n-t+1}(H)+(2^{k}-2)\prod\limits_{t=1}^{k}\left[\lambda_{i_{t}}(F)\lambda_{n-t+1}(H)\right]^{\frac{1}{2}}$
$\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\left[1-\lambda_{n-i_{t}+1}(B^{*}B)\right]+\prod\limits_{t=1}^{k}\left[1-\lambda_{n-t+1}(A^{*}A)\right]^{-1}\sigma^{2}_{n-t+1}(A-B)$
$\displaystyle+(2^{k}-2)\prod\limits_{t=1}^{k}\left[\lambda_{i_{t}}(F)\lambda_{n-t+1}(H)\right]^{\frac{1}{2}}.$

Combining (4.11) and (4.12), we obtain the desired inequalities. ∎

Setting $k=n$ in Theorem 4.3 gives a stronger version of the Hua’s inequality.

######  Corollary 4.1.

Let $A,B\in\mathcal{SC}_{n\times n}$. Then

$\displaystyle\left|\det\left(I-A^{*}B\right)\right|^{2}\geq\det\left(I-A^{*}A\right)\det\left(I-B^{*}B\right)+\left|\det\left(A-B\right)\right|^{2}$
$\displaystyle\qquad\qquad\qquad\qquad+(2^{n}-2)\big{[}\det\left(I-A^{*}A\right)\det\left(I-B^{*}B\right)\big{]}^{\frac{1}{2}}\left|\det\left(A-B\right)\right|.$

Below is a reversal inequality of the previous theorem.

######  Theorem 4.4.

Let $A,B\in\mathbb{C}^{n\times n}$. Then

$\displaystyle\prod\limits_{t=1}^{k}\sigma^{2}_{i_{t}}\big{(}I-A^{*}B\big{)}$
$\displaystyle\leq$
$\displaystyle\prod\limits_{t=1}^{k}\big{[}1+\lambda_{i_{t}}(A^{*}A)\big{]}\big{[}1+\lambda_{t}(B^{*}B)\big{]}-\prod\limits_{t=1}^{k}\sigma^{2}_{n-t+1}\big{(}A+B\big{)}$
(4.13)
$\displaystyle-(2^{k}-2)\prod\limits_{t=1}^{k}\sigma_{i_{t}}\big{(}I-A^{*}B\big{)}\sigma_{n-t+1}\big{(}A+B\big{)}.$

###### Proof.

For $A,B\in\mathbb{C}^{n\times n}$, the following matrix identity holds (see,
e.g., [23, pp. 228]):

$I+A^{*}A=P+Q$

where

$P=(A+B)^{*}(I+BB^{*})^{-1}(A+B),$
$Q=(I-A^{*}B)(I+B^{*}B)^{-1}(I-A^{*}B)^{*}.$

By Lemma 2.2, we have

$\displaystyle\prod\limits_{t=1}^{k}\lambda_{n-t+1}(P)$
$\displaystyle\geq\prod\limits_{t=1}^{k}\big{[}1+\lambda_{t}(BB^{*})\big{]}^{-1}\sigma^{2}_{n-t+1}\big{(}A+B\big{)},$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{i_{t}}(Q)$
$\displaystyle\geq\prod\limits_{t=1}^{k}\big{[}1+\lambda_{t}(B^{*}B)\big{]}^{-1}\sigma^{2}_{i_{t}}\big{(}I-A^{*}B\big{)}.$
(4.14)

Using (3.3) in Theorem 3.1, we derive

$\displaystyle\prod\limits_{t=1}^{k}\big{[}1+\lambda_{i_{t}}(A^{*}A)\big{]}=\prod\limits_{t=1}^{k}\lambda_{i_{t}}(P+Q))$
$\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\lambda_{n-t+1}(P)+\prod\limits_{t=1}^{k}\lambda_{i_{t}}(Q)+(2^{k}-2)\prod\limits_{t=1}^{k}[\lambda_{n-t+1}(P)\lambda_{i_{t}}(Q)]^{\frac{1}{2}}$
$\displaystyle\geq$
$\displaystyle\prod\limits_{t=1}^{k}\big{[}1+\lambda_{t}(B^{*}B)\big{]}^{-1}\bigg{[}\prod\limits_{t=1}^{k}\sigma^{2}_{i_{t}}\big{(}I-A^{*}B\big{)}+\prod\limits_{t=1}^{k}\sigma^{2}_{n-t+1}\big{(}A+B\big{)}$
$\displaystyle+(2^{k}-2)\prod\limits_{t=1}^{k}\sigma_{i_{t}}\big{(}I-A^{*}B\big{)}\sigma_{n-t+1}\big{(}A+B\big{)}\bigg{]}.$

Multiplying by $\prod\limits_{t=1}^{k}\big{[}1+\lambda_{t}(B^{*}B)\big{]}$, we
obtain the desired inequalities (4.13). ∎

As (4.3) is a reversal of Hua’s (4.1), the following result, as a special case
of the Theorem 4.4 by setting $k=n$, may be viewed as a counterpart of
Corollary 4.1.

######  Corollary 4.2.

Let $A,B\in\mathbb{C}^{n\times n}$. Then

$\displaystyle\big{|}\det\left(I-A^{*}B\right)\big{|}^{2}$ $\displaystyle\leq$
$\displaystyle\det\big{(}I+A^{*}A\big{)}\det\left(I+B^{*}B\right)-\big{|}\det\left(A+B\right)\big{|}^{2}$
$\displaystyle-(2^{n}-2)\big{|}\det\left(I-A^{*}B\right)\big{|}\big{|}\det\left(A+B\right)\big{|}.$

Consequently,

$\displaystyle\big{|}\det\left(I-A^{*}B\right)\big{|}^{2}$
$\displaystyle\leq\det\left(I+A^{*}A\right)\det\left(I+B^{*}B\right)-\big{|}\det\left(A+B\right)\big{|}^{2}.$

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# Centre Vortices in the Presence of Dynamical Fermions

James Biddle  Waseem Kamleh  Derek Leinweber

###### Abstract

The behaviour of centre vortices in the presence of dynamical fermions is
studied for the first time in the context of the static quark potential and
the Landau-gauge gluon propagator. These results indicate that in the presence
of dynamical fermions, centre vortices are able to better encapsulate the
long-range behaviour of QCD, compared to previous studies on pure Yang-Mills
lattices. This work provides strong evidence that centre vortices are
responsible for the long-range linear potential and the enhanced infrared
gluon propagator, which are indicative of confinement in QCD.

##  1 Introduction

There is now an extensive body of lattice QCD results showing that centre
vortices [1, 2] play an important role in the confinement of quarks. Previous
work has demonstrated that removal of centre vortices results in the loss of
dynamical mass generation and restoration of chiral symmetry [3, 4], removal
of the string tension [5, 6] and suppression of the infrared Landau-gauge
gluon propagator [7, 6, 8]. It has also been shown that vortices alone can
reproduce a linear static quark potential [5, 9, 10] and the infrared
enhancement of the Landau-gauge gluon propagator. These results speak to the
deep ties between centre vortices and confinement. However, the value of the
vortex-only string tension obtained from pure Yang-Mills lattice studies has
been consistently shown to be about $\sim 62\%$ of the full string tension.
Additionally, upon removal of centre vortices the gluon propagator shows
persistent infrared enhancement [8], indicating that there are still residual
non-perturbative effects present.

In this work, we perform the first calculation exploring the impact of
dynamical fermions on centre vortices as they govern the static quark
potential and the Landau-gauge gluon propagator. By isolating the centre-
vortex contributions to these quantities, we investigate how dynamical
fermions impact the centre vortex model ground-state field structure. This
calculation reveals significant new insight into the role of centre vortices
in our understanding of QCD vacuum structure.

##  2 Method

The centre vortex picture describes the QCD vacuum via the group centre of
$SU(3)$, given by the cube roots of unity. Within this picture, centre
vortices appear in four dimensions as closed sheets percolating the vacuum. As
the sheets pierce a three dimensional slice of the lattice, they create a
vortex line. Visualisations of centre vortices on the lattice are presented in
Ref. [11]. The key property is that as a vortex line pierces a planar Wilson
loop, it contributes a non-trivial centre phase to the loop, namely one of
$z=\exp\left(\pm\frac{2\pi i}{3}\right)$. From this property, it is possible
to identify centre vortices on the lattice [5, 10]. This identification
procedure leads us to define three ‘vortex-modified’ ensembles:

  1. 1.

Original, untouched (UT) fields, $U_{\mu}(x)$,

  2. 2.

Vortex-only (VO) fields, $Z_{\mu}(x)~{}\in~{}\exp\left(\frac{2\pi
i}{3}m(x)\right),~{}m(x)\in\left\\{-1,\,0,\,+1\right\\}$,

  3. 3.

Vortex-removed (VR) fields, $Z_{\mu}(x)^{\dagger}\,U_{\mu}(x)=R_{\mu}(x)$,

This procedure is performed on three gauge field ensembles, each containing
200 configurations. Two of these are $(2+1)$ flavour dynamical ensembles from
the PACS-CS collaboration [12]. The remaining ensemble is pure-gauge with a
lattice spacing tuned to be similar to that of the PACS-CS ensembles. The
pure-gauge ensemble provides a point of reference with which we can evaluate
the impact of dynamical fermions on centre-vortex phenomenology.

##  3 Static Quark Potential

The static quark potential is obtained by considering the expectation value of
Wilson loops, $\left\langle W(r,t)\right\rangle$, with spatial separation $r$
and temporal extent $t$ [13],

$\langle
W(r,\,t)\rangle=\sum_{\alpha}\lambda^{\alpha}\\!(r)\,\exp\left(-V^{\alpha}\\!(r)\,t\right)\,.$
(1)

The relevant static quark potential is given by the lowest $\alpha=0$ state,
which we extract via a variational analysis.

We expect the static quark potential for the original configurations to follow
a Cornell potential, such that

$V(r)=V_{0}-\frac{\alpha}{r}+\sigma\,r$ (2)

The Cornell potential naturally decomposes into a Coulomb term and a linear
term, with the former dominating at short distances and the latter at large
distances. As we expect that centre vortices will encapsulate the non-
perturbative long-range physics, the vortex-only results should give rise to a
linear potential [14, 15, 16], whereas the vortex-removed results should
capture the short-range behaviour. Initially we calculate the static quark
potential on the pure-gauge ensemble, making use of our three vortex-modified
ensembles. To analyse the linearity of the potential at large distances we
also plot a sliding linear fit to the potential with extent $-\frac{3}{2}a\leq
r\leq\frac{3}{2}a$. These results are shown in Fig. 1.

Figure 1:  The static quark potential as calculated on the vortex-modified
pure-gauge ensembles. The lower plot shows the slope of a sliding linear
window, as described in the text.

We observe that qualitatively the centre vortices account for the long-
distance physics, as can be seen in the linearity of the vortex-only data.
Furthermore, removal of vortices accounts for a complete removal of long-range
potential, as visualised by the sliding fits dropping to 0. However, the value
of the string tension between the original and vortex-only data differs. As
has been observed previously, the vortex-only result sits at around $62\%$ of
the original string tension.

We now introduce dynamical fermions with a pion mass of
$m_{\pi}=701$\mathrm{M}\mathrm{e}\mathrm{V}$$ and perform the same analysis.
The results of this are shown in Fig. 2. Here we observe a slight suppression
of the original string tension, as expected due to screening effects brought
about by the introduction of dynamical fermions. Most importantly, the vortex-
only string tension is now in excellent agreement with the original ensemble,
as can be clearly seen on the sliding fit plot. The fact that this improvement
is observed even at an unphysically large pion mass is interesting, as it
speaks to the dramatic effect of dynamical fermions on the vortex structure
even when the observed screening effects are minimal. The effect of vortex
removal is to remove confinement, with the sliding average in excellent
agreement with 0 at large distances. Similar results are observed for the
lightest pion mass available in the PACS-CS ensembles at
$m_{\pi}=156$\mathrm{M}\mathrm{e}\mathrm{V}$$.

Figure 2:  The static quark potential as calculated on the vortex-modified
$m_{\pi}=701$\mathrm{M}\mathrm{e}\mathrm{V}$$ ensembles.

Overall, the introduction of dynamical fermions both improves the reproduction
of the string tension from vortex-only ensembles and improves the removal of
long-range potential from vortex-removed ensembles. These results are
surprising, and demand further investigation into the interplay between centre
vortices and dynamical fermions.

##  4 Gluon Propagator

We now turn our attention to the Landau-gauge gluon propagator in momentum
space, defined on the lattice as

$D(q^{2})=\frac{2}{3\,(n_{c}^{2}-1)\,V}\left\langle\operatorname{Tr}A_{\mu}(q)\,A_{\mu}(-q)\right\rangle\,,$
(3)

where $V$ is the lattice volume and $n_{c}=3$ is the number of colours. The
nonperturbative gluon propagator can be written as

$D(q^{2})=\frac{Z(q^{2})}{q^{2}}\,,$ (4)

where $Z(q^{2})$ is the renormalisation function. In the limit as
$q\rightarrow\infty$, we expect $D(q^{2})\rightarrow\frac{1}{q^{2}}$, and as
such we plot $Z(q^{2})$. $Z(q^{2})$ is renormalised by setting $Z(\mu^{2})=1$
at $\mu=5.5$\mathrm{G}\mathrm{e}\mathrm{V}$$.

Beginning with the pure-gauge ensemble, the vortex-modified propagators are as
shown in Fig. 3. The original propagator is epitomised by infrared enhancement
and a trend towards unity at large momentum. Here we see that vortex removal
suppresses the infrared enhancement, but retains the high momentum behaviour.
Conversely, the vortex-only propagator contains only infrared strength, with a
complete removal of high momentum behaviour. We also consider reconstructing
the original propagator as a linear combination of the vortex-only and vortex-
removed propagators. Here we see that we can achieve a full reproduction of
the original propagator via this method, indicating that vortices effectively
partition the propagator short and long-range information. However, there is
still substantial residual infrared strength in the vortex-removed propagator.

Figure 3:  Pure-gauge vortex-modified gluon propagators.

Turning to the dynamical ensemble with
$m_{\pi}=701$\mathrm{M}\mathrm{e}\mathrm{V}$$, the vortex-modified propagators
now appear as in Fig. 4. We see a noticeable screening of the propagator,
coupled with improvement in the ability of vortex-removal to remove infrared
enhancement. The ability to reconstruct the original propagator from its
vortex components is diminished, and at present the origins of this
discrepancy are under investigation. However, the increased suppression of
infrared strength after vortex removal is an important point, as it indicates
a substantial enhancement in the isolation of infrared strength into the
vortex-only propagator.

Figure 4:  $m_{\pi}=701$\mathrm{M}\mathrm{e}\mathrm{V}$$ vortex-modified gluon
propagators.

##  5 Conclusion

We have presented the first study of the impact of dynamical fermions on
centre vortices in the context of the static quark potential and the gluon
propagator. In regards to the static quark potential, we find that centre
vortices can quantitatively capture the string tension, and that vortex
removal eliminates the long distance potential. This is in stark contrast to
the pure-gauge situation, where vortices only qualitatively reproduce the
long-range potential.

When considering the gluon propagator, the results are encouraging. The
increased ability of vortex removal to suppress infrared enhancement is an
important result that demands further investigation. The discrepancy in
reconstructing the original propagator from its vortex-modified components is
also being studied further.

These early results are exciting and perhaps unexpected. The results presented
here show a significant change in the ability of centre vortices to capture
the underlying physics of confinement. Further calculation is needed to
clarify this picture, but so far these results point to a host of exciting
relationships between centre vortices and dynamical fermions on the lattice.

## Acknowledgments

We thank the PACS-CS Collaboration for making their 2 +1 flavour
configurations available via the International Lattice Data Grid (ILDG). This
research was undertaken with the assistance of resources from the National
Computational Infrastructure (NCI), provided through the National
Computational Merit Allocation Scheme and supported by the Australian
Government through Grant No. LE190100021 via the University of Adelaide
Partner Share. This research is supported by Australian Research Council
through Grants No. DP190102215 and DP210103706. WK is supported by the Pawsey
Supercomputing Centre through the Pawsey Centre for Extreme Scale Readiness
(PaCER) program.

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