# Catoni-style Confidence Sequences under Infinite Variance

Sujay Bhatt, Guanhua Fang, Ping Li
Cognitive Computing Lab, Baidu Research
10900 NE 8th St. Bellevue, WA 98004 &Gennady Samorodnitsky
School of ORIE, Cornell University
220 Frank T Rhodes Hall, Ithaca, NY 14853

###### Abstract

In this paper, we provide an extension of confidence sequences for settings
where the variance of the data-generating distribution does not exist or is
infinite. Confidence sequences furnish confidence intervals that are valid at
arbitrary data-dependent stopping times, naturally having a wide range of
applications. We first establish a lower bound for the width of the Catoni-
style confidence sequences for the finite variance case to highlight the
looseness of the existing results. Next, we derive tight Catoni-style
confidence sequences for data distributions having a relaxed bounded
$p^{th}-$moment, where $p\in(1,2]$, and strengthen the results for the finite
variance case of $p=2$. The derived results are shown to better than
confidence sequences obtained using Dubins-Savage inequality.

##  1 Introduction

Sequential design of experiments is a classical framework in statistical
sampling theory, in which the size and the composition of samples are not
fixed in advance and are allowed to be functions of the observations
themselves [13]. Confidence sequence is one particular tool in sequential
design that facilitates anytime-valid inference [3, 6, 5]. In particular,
confidence sequence is a sequence of confidence intervals that is valid at
data-dependent stopping times.

Formally, let $X_{1},X_{2},\cdots$ be an i.i.d stream drawn from distribution
$P$. The basic object of interest is the unknown mean of this distribution,
namely, $\mu=\mathbf{E}_{P}[X_{1}]$. A crude way to quantify the uncertainty
associated with the mean estimation problem is via confidence intervals. Here
one constructs a $\Sigma(X_{1},\cdots,X_{t})-$measurable interval
$\texttt{CI}_{t}$ for each $t\in\mathbb{N}^{+}$ such that
$\forall~{}t\in\mathbb{N}^{+}$, the following holds:
$\mathbb{P}(\mu\in\texttt{CI}_{t})\geq 1-\alpha$, for some coverage
probability $1-\alpha\in(0,1)$. However, as argued in [14], confidence
intervals may undercover at stopping times. Also, it is well known that the
data-dependent peeking at confidence intervals inflates the Type-1 error [8,
5]. This motivates confidence sequences, which provide an universal
quantification over $t$. For a confidence parameter $\alpha\in(0,1)$ and
$t\in\mathbb{N}^{+}$, the sequence of random intervals $\\{\texttt{CI}_{t}\\}$
that satisfy
$\mathbb{P}(\forall~{}t\in\mathbb{N}^{+},~{}\mu\in\texttt{CI}_{t})\geq
1-\alpha$ are called a $(1-\alpha)-$confidence sequence.

Literature. Confidence sequences (CS) are instrumental in modern application
tools like multi-armed bandits [17], A/B testing [5], causal inference [12],
etc. Given the importance, it is not surprising that a significant research
effort has been allocated to construct confidence sequences under various
distributional assumptions on the data. [3, 7] consider $P$ as normal
distributed and construct confidence sequences, while [9, 8] consider $P$
belonging to an exponential family. [15] consider arbitrary but bounded $P$,
while [5] consider $P$ having a bounded moment generating function.

Recently, [14] make a substantial contribution to the literature on confidence
sequences by relaxing the distributional assumptions to requiring only the
existence of a bounded second moment. This is made possible by using robust
mean estimator developed in [1], which uses influence functions to stabilize
the effect of the outliers. While the previous works [5, 15] required a
Chernoff-type assumption on the distribution resulting in $O(\sqrt{\log t/t})$
shrinkage rates for the confidence sequences, [14] show that employing
Catoni’s estimator improves the rate to $O(\sqrt{\log\log 2t/t})$ under weaker
assumptions on the distribution. The significance of this result is that there
is no excess compromise in considering confidence sequences or weakening the
distributional assumptions. A careful reading of [14] shows however that there
are inefficiencies compared to the lower bound that can be improved. Also,
[14] cannot deal with situations where even the second moment does not exist.

The key improvements over the existing results are as follows:

  * •

We extend the analysis to the case of $p\in(1,2)$, thereby relaxing the
distributional assumptions on the data. This allows sequential design for a
larger class of distributions, while achieving good control over the width of
the CS.

  * •

We then derive lower bounds for the width of Catoni-style confidence sequences
(CS) for the finite variance case. This shows that [14] almost - but not quite
\- matches the lower bound, leaving room for improvement.

  * •

We improve on the näive CS derived using generalized Dubins-Savage inequality
in terms of the growth of the width of the confidence interval (CI) w.r.t the
confidence parameter $\alpha$. We improve on the Catoni-style CS in [14] by
obtaining a smaller CI width.

Note that we consider i.i.d data as it has much wider applicability and has
simpler notation, however, the results carryover for stochastic processes with
constant conditional expectation using the influence function and the
supermartingale arguments as in [14].

Organization. The paper is organized as follows. Sec. 2 formulates the problem
and the key objectives that we seek to address. Sec. 3 presents a first
attempt at deriving confidence sequences using the well known Dubins-Savage
inequality for $p\in(1,2]$. Sec. 4 derives the lower bound for the width of
the confidence sequences when the data has finite variance. Sec. 5 presents
the improvement for the finite variance case and the generalization to the
infinite variance case.

##  2 Problem Formulation

In this section, we first introduce the notation and describe the main problem
considered in this paper. Recall that $X_{1},X_{2},\cdots$ are a sequence of
random variables with mean $\mathbb{E}[X_{1}]=\mu$ and the $p^{th}-$moment
$\mathbb{E}|X_{1}-\mu|^{p}\leq\upsilon_{p}$ for $p\in(1,2]$. A level
$1-\alpha\in(0,1)$ confidence sequence for $\mu$ is a sequence of real numbers
$L_{n}(X_{1},\cdots,X_{n})$ and $U_{n}(X_{1},\cdots,X_{n})$, where
$L_{n},U_{n}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ with $n=1,2,\cdots$ such
that $L_{n}\leq U_{n}$ point-wise and

$\mathbb{P}\Big{(}L_{n}(X_{1},\cdots,X_{n})\leq\mu\leq
U_{n}(X_{1},\cdots,X_{n}),\forall n\geq 1\Big{)}\geq 1-\alpha.$

Let the width of the $n^{th}$ confidence interval

$W_{n}(X_{1},\cdots,X_{n}):=U_{n}(X_{1},\cdots,X_{n})-L_{n}(X_{1},\cdots,X_{n}).$

Objective. We are interested in how fast this width $W_{n}$ can shrink as $n$
increases, when the $p^{th}-$moment of the data distribution $P$ is bounded
with $p\in(1,2]$.

##  3 Confidence Sequence for Infinite Variance via Dubins-Savage

In this section, we take the first steps to derive confidence sequences for
the infinite variance case. We borrow the width optimization ideas from [15,
14] and make use of the $L_{p}$ version of the classical Dubins-Savage. A
generalization of the classical Dubins-Savage inequality [4] first appeared in
[10], and most recently in [11]. Lemma 1 is a restatement and can be derived
using an approach of Doob [11].

###### Lemma 1

Let $\\{X_{t}\\}$ be a real-valued stochastic process adapted to the
filtration $\\{\mathcal{F}_{t}\\}$, where $\mathcal{F}_{0}$ is the trivial
sigma-algebra. Let $\\{S_{t}\\}$ be a martingale with
$V_{t}=\mathbb{E}[|S_{t}-S_{t-1}|^{p}|\mathcal{F}_{t-1}]$. Then for all $a\geq
0,~{}b>0$, we have $\mathbb{P}\Big{(}S_{t}\geq
a+b\sum_{i=1}^{t}V_{i}\Big{)}\leq\frac{1}{(1+m_{p}ab^{\frac{1}{p-1}})^{p-1}},$
where $m_{p}=\Big{(}\frac{p-1}{2^{2-p}}\Big{)}^{\frac{1}{p-1}}$.

We have the following result for the confidence sequence.

######  Theorem 1 (Dubins-Savage)

Let
$a=\frac{1}{m_{p}b^{\frac{1}{p-1}}}\Big{(}\Big{(}\frac{2}{\alpha}\Big{)}^{\frac{1}{p-1}}-1\Big{)}$.
The width of the confidence interval using Dubins-Savage inequality is given
as

$W_{n}(X_{1},\cdots,X_{n})=\frac{2b\upsilon_{p}\sum_{i=1}^{n}\lambda_{i}^{p}}{\sum_{i=1}^{n}\lambda_{i}},$

where the coefficients $\lambda_{i}$ are chosen as
$\lambda_{t}=\Big{(}\frac{a}{tb\upsilon_{p}(p-1)}\Big{)}^{1/p}$ for $t\leq n$.

From Theorem 1, the width of the confidence interval shrinks at the rate
$O(\frac{\log t}{t^{\frac{p-1}{p}}})$. When $p=2$, we obtain
$\widetilde{O}(t^{-1/2})$ which is known to be unimprovable [5]. The
dependence on $\alpha$ is $O(\alpha^{-1/p})$.

##  4 Lower Bound for Finite Variance ($p=2$)

In this section, we establish the lower bounds for the width of Catoni-style
confidence sequences by making similar assumptions as in [14]. We make use of
the general law of iterated logarithm by [16] to support key arguments.

[14] discuss one possible way to construct a Catoni-style confidence sequence
and is as follows. Suppose the observations $X_{i}$ have a finite second
moment and $\texttt{Var}(X_{1})\leq\sigma^{2}$ for a known $\sigma^{2}$.
Consider an increasing function $\psi$ such that for $x\in\mathbb{R}$ as in
[1],

$\psi(x)=\begin{cases}-\log(1-x+x^{2}/2),~{}x<0\\\ \log(1+x+x^{2}/2),~{}x\geq
0.\end{cases}$

Let $(\lambda_{i})$ denote a sequence of positive numbers. Choose positive
sequences $(a_{n})$ and $(b_{n})$ with $a_{n}\leq b_{n}$ for each $n$ and
define

$\displaystyle L_{n}(X_{1},\cdots,X_{n})$ $\displaystyle:=\text{solution of
the equation}~{}\sum_{i=1}^{n}\psi(\lambda_{i}(X_{i}-x))=b_{n},$
$\displaystyle U_{n}(X_{1},\cdots,X_{n})$ $\displaystyle:=\text{solution of
the equation}~{}\sum_{i=1}^{n}\psi(\lambda_{i}(X_{i}-x))=a_{n}.$ (1)

[14, Theorem 9 & 10] shows that, for a specific choice of the sequences
$(a_{n})$ and $(b_{n})$, 4 defines a confidence sequence for $\mu$.
Furthermore, under certain conditions on the sequence $(\lambda_{i})$, for all
$n$ large enough,

$W_{n}(X_{1},\cdots,X_{n})\leq
4\frac{\sigma^{2}\sum_{i=1}^{n}\lambda_{i}^{2}+G(\alpha,\delta)}{\sum_{i=1}^{n}\lambda_{i}}$
(2)

for some $n-$independent constant $G(\alpha,\delta)$ and $\delta\in(0,1)$. We
are interested in establishing a lower bound nearly matching (2). We will
allow any choice of the sequences $(a_{n})$ and $(b_{n})$ and a larger
flexibility in the choice of the function $\psi$.

###### Theorem 2

Let $\texttt{Var}(X_{1})=\sigma^{2}$ and
$\mathbb{E}|X_{1}|^{2+\vartheta}<\infty$ for some $0<\vartheta\leq 1$. Let the
sequence $(\lambda_{i})$ be such that $\lambda_{i}\downarrow 0$ and
$\sum_{i=1}^{\infty}\lambda_{i}^{2}=\infty$. Let the influence function $\psi$
be $1-$Lipschitz. Then, the width of the confidence sequence
$W_{n}(X_{1},\cdots,X_{n})$ is such that

$W_{n}(X_{1},\cdots,X_{n})\geq\frac{a\Big{(}\sum_{i=1}^{n}\lambda_{i}^{2}\log\log\sum_{i=1}^{n}\lambda_{i}^{2}\Big{)}^{1/2}}{\sum_{i=1}^{n}\lambda_{i}},$

for any $a<2\sigma\sqrt{2}$ and at least for large $n$.

The result provides the minimum width of the confidence sequence in the finite
variance case. Below, we sketch the broad ideas employed in establishing the
result. Notice that since the influence function $\psi$ is $1-$Lipschitz, we
have
$\Big{|}\frac{d}{dx}\sum_{i=1}^{n}\psi(\lambda_{i}(X_{i}-x))\Big{|}\leq\sum_{i=1}^{n}\lambda_{i}.$
Therefore, the confidence intervals defined in (4) satisfy

$W_{n}(X_{1},\cdots,X_{n})\geq\frac{b_{n}-a_{n}}{\sum_{i=1}^{n}\lambda_{i}}.$

Next, if (4) defines the confidence sequence for $\mu$ we need to show that
$b_{n}-a_{n}$ cannot be too small. Indeed, from (4) we have

$\mathbb{P}\Big{(}a_{n}\leq\sum_{i=1}^{n}\psi(\lambda_{i}(X_{i}-m))\leq
b_{n}~{}\text{for all}~{}n\geq 1\Big{)}\geq 1-\alpha.$ (3)

Consider the transformation $Y_{i}=\psi(\lambda_{i}(X_{i}-m))$, whence
$Y_{1},Y_{2},\cdots$ is a sequence of independent random variables with all
finite moments. For $n\geq 1$, let
$\widetilde{a}_{n}=a_{n}-\sum_{i=1}^{n}\mathbb{E}Y_{i},~{}~{}\widetilde{b}_{n}=b_{n}-\sum_{i=1}^{n}\mathbb{E}Y_{i}.$
Then (3) implies that

$\mathbb{P}\Big{(}\widetilde{a}_{n}\leq\sum_{i=1}^{n}(Y_{i}-\mathbb{E}Y_{i})\leq\widetilde{b}_{n}~{}\text{for
all}~{}n\geq 1\Big{)}\geq 1-\alpha.$ (4)

Clearly, $\widetilde{b}_{n}-\widetilde{a}_{n}=b_{n}-a_{n}$, and we will show
that these differences cannot be too small using (3) and the general law of
iterated logarithm in [16].

##  5 Catoni-style Confidence Sequence for $p\in(1,2]$

From Theorem 2, it is clear that the lower bounds are sharper than the width
obtained (see (2)) in [14]. In this section, we extend the analysis and
establish the width of the confidence interval for $p=(1,2]$, and strengthen
the result in [14] for $p=2$. Note that the constant $G(\alpha,\delta)$ in (2)
depends on an additional parameter $\delta\in(0,1)$, with the width holding
with probability $1-\delta$. We both extend the approach in [14] to the case
of infinite variance and strengthen their result, even in the case of finite
variance, where the constant is only a function of $\alpha$.

For an appropriate $C_{p}>0$ let $\psi$ be a non-decreasing function
$\mathbb{R}\to\mathbb{R}$ such that

$-\log(1-x+C_{p}|x|^{p})\leq\phi(x)\leq\log(1+x+C_{p}|x|^{p})$ (5)

for all $x\in\mathbb{R}$. One way to chose
$C_{p}=\Big{(}\frac{p-1}{p}\Big{)}^{p/2}\Big{(}\frac{2-p}{p-1}\Big{)}^{(2-p)/2}$,
whence $C_{2}=1/2$ as in [1]. Let $(\lambda_{n})$ be a sequence of positive
numbers such that

$\lim_{n\to\infty}\lambda_{n}=0,\ \ \
\sum_{n=1}^{\infty}\lambda_{n}^{p}=\infty.$ (6)

It follows immediately that the processes
$M_{n}^{+}=\prod_{i=1}^{n}\exp\bigl{\\{}\phi\bigl{(}\lambda_{i}(X_{i}-\mu)\bigr{)}-C_{p}v_{p}\lambda_{i}^{p}\bigr{\\}}$
and
$M_{n}^{-}=\prod_{i=1}^{n}\exp\bigl{\\{}-\phi\bigl{(}\lambda_{i}(X_{i}-\mu)\bigr{)}-C_{p}v_{p}\lambda_{i}^{p}\bigr{\\}}$
are non-negative supermartingales with respect to the natural filtration of
the sequence $X_{1},X_{2},\ldots$. Let $0<\alpha<1$ be a confidence level. By
the maximal inequality for non-negative supermartingales, the following
sequence of sets forms a $(1-\alpha)$-confidence sequence for $\mu$:

$\displaystyle
I_{n}(\alpha)=\left\\{x\in\mathbb{R}:\,-\log\frac{2}{\alpha}-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\leq\sum_{i=1}^{n}\phi\bigl{(}\lambda_{i}(X_{i}-x)\bigr{)}\leq\log\frac{2}{\alpha}+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\right\\},$
(7)

$n=1,2,\ldots$. Note that by (6), the sum $\sum_{i=1}^{n}\lambda_{i}^{p}$
grows slower than linearly fast with $n$. Therefore, at least for large $n$
the equations

$\sum_{i=1}^{n}\phi\bigl{(}\lambda_{i}(X_{i}-x)\bigr{)}=\pm\left(\log\frac{2}{\alpha}+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\right)$
(8)

has unique real roots, which we will denote by $x_{+,n}$ and $x_{-,n}$
respectively, in which case the set $I_{n}(\alpha)$ is an interval of the
finite length $\bigl{|}I_{n}(\alpha)\bigr{|}=x_{-,n}-x_{+,n}.$ The next result
characterizes how fast these lengths grow as $n$ increases.

###### Theorem 3

Suppose the sequence $(\lambda_{n})$ is non-random, $0<t_{n}<1$ and
$\tau_{n}>0$. Suppose
$\varepsilon_{n}=\alpha\exp\Big{\\{}-C_{p}\upsilon_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1+t_{i})^{-(p-1)}\Big{\\}}$
for $n=1,2,\cdots$. Consider the condition

$\displaystyle C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1+t_{i}^{-(p-1)})+\log
2/\alpha+\log
2/\varepsilon_{n}\leq\frac{\tau_{n}^{1/(p-1)}}{(1+\tau_{n})^{p/(p-1)}}\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(C_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right)^{1/(p-1)}}.$
(9)

The following holds for the width of the confidence sequence

$P\biggl{(}\bigl{|}I_{n}(\alpha)\bigr{|}\leq
4(1+\tau_{n})\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\alpha}{\sum_{i=1}^{n}\lambda_{i}},\text{~{}$\forall n$ such
that~{}\eqref{eq:cond_mr} holds}\biggr{)}\geq
1-\alpha\sum_{i=1}^{\infty}\varepsilon_{i}.$

Note that with $\varepsilon_{n}$ chosen as in the statement, the condition (9)
holds for all large $n$, at least if $(t_{n})$ are bounded away from 0, and
$(\tau_{n})$ are not too small (we could, in fact, keep $\tau_{n}$ a small
positive constant). The right hand side can be made small if $\alpha$ is
small.

Note that $W_{n}(X_{1},\cdots,X_{n}):=\bigl{|}I_{n}(\alpha)\bigr{|}$, so
comparing with (2) for $p=2$, we note the following differences: (i) The width
in (2) depends on both $\alpha$ and another confidence parameter $\delta$
implying a compromise in the width, while the width in Theorem 3 depends only
on the confidence parameter $\alpha$. (ii) Abusing the notation, let
$W_{n,\lambda}=W_{n}(X_{1},\cdots,X_{n})\sum_{i=1}^{n}\lambda_{i}$. For $p=2$,
using Theorem 3 $W_{n,\lambda}\leq
2(1+\tau_{n})\sigma^{2}\sum_{i=1}^{n}\lambda_{i}^{2}(1+1/t_{i})+\log(2/\delta)$,
while that from (2) is $W_{n,\lambda}\leq
4\sigma^{2}\sum_{i=1}^{n}\lambda_{i}^{2}+\log(2/\delta)+\log(2/\alpha)$;
implying sharper bounds from Theorem 3.

Also, $\lambda_{i}=\Theta(1/t^{1/p})$ from [1, 2] implying that the Catoni-
style confidence sequence enjoys $O(\frac{\log t}{t^{\frac{p-1}{p}}})$
shrinkage rate. The dependence on $\alpha$ is $O(\log(1/\alpha))$ improving
over Theorem 1, as the width increases slowly in case of Catoni-style sequence
as $\alpha\downarrow 0$.

##  6 Proofs of Main Results

In this section, we provide proofs of the main results to illustrate the
challenges involved.

Proof of Theorem 1: Let $S^{+}_{n}=\sum_{i=1}^{n}\lambda_{i}(X_{i}-\mu)$ and
$S^{-}_{n}=\sum_{i=1}^{n}-\lambda_{i}(X_{i}-\mu)$ denote two martingales. The
confidence intervals and hence the sequence is obtained by applying Lemma 1 to
each of these martingales. Let
$a=\frac{1}{m_{p}b^{\frac{1}{p-1}}}\Big{(}\Big{(}\frac{2}{\alpha}\Big{)}^{\frac{1}{p-1}}-1\Big{)}$.
We have from Lemma 1,

$\displaystyle\mathbb{P}\Bigg{(}\forall~{}n,\sum_{i=1}^{n}\lambda_{i}(X_{i}-\mu)\leq
a+b\sum_{i=1}^{n}\lambda_{i}^{2}\mathbb{E}[|X_{i}-\mu|^{p}|\mathcal{F}_{n-1}]\Bigg{)}$
$\displaystyle\geq 1-\alpha/2,$
$\displaystyle\mathbb{P}\Bigg{(}\forall~{}n,-\sum_{i=1}^{n}\lambda_{i}(X_{i}-\mu)\leq
a+b\sum_{i=1}^{n}\lambda_{i}^{2}\mathbb{E}[|X_{i}-\mu|^{p}|\mathcal{F}_{n-1}]\Bigg{)}$
$\displaystyle\geq 1-\alpha/2.$

By using the fact that
$\mathbb{E}[|X_{i}-\mu|^{p}|\mathcal{F}_{n-1}]\leq\upsilon_{p}$ and taking an
union bound the result follows. The sequence that optimizes the width is
calculated using [15, Eq. (24-28)] and [14, Appendix A] as the minimizer of
$b\upsilon_{p}\lambda^{p-1}+\frac{a}{t\lambda},$ solving which we obtain the
desired sequence.

Proof of Theorem 2: Let $s^{2}_{n}=\sum_{i=1}^{n}\texttt{Var}(Y_{i})$ for
$n=1,2,\cdots$. We will first show that
$\texttt{Var}(Y_{i})\sim\lambda_{i}^{2}\sigma^{2}$ as $i\rightarrow\infty$.
Indeed,
$\mathbb{E}Y_{i}=\mathbb{E}\psi(\lambda_{i}(X-\mu))\leq\mathbb{E}[\log(1+\lambda_{i}(X-\mu))+\lambda_{i}^{2}(X-\mu)^{2}/2]\leq\mathbb{E}[\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2}/2]=\lambda_{i}^{2}\sigma^{2}/2$.
Also,
$\mathbb{E}Y_{i}\geq\mathbb{E}[-\log(1-\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]\geq\mathbb{E}[-(-\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2}/2)]=-\lambda_{i}^{2}\sigma^{2}/2.$
Therefore, $|\mathbb{E}Y_{i}|\leq\lambda_{i}^{2}\sigma^{2}/2$ for
$i=1,2,\cdots$. Next,

$\displaystyle\mathbb{E}Y_{i}^{2}=\mathbb{E}\psi^{2}(\lambda_{i}(X-\mu))$
$\displaystyle\leq\mathbb{E}\Big{\\{}[\log(1+\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]^{2}\mathds{1}(X\geq\mu)\Big{\\}}$
$\displaystyle+\mathbb{E}\Big{\\{}[\log(1-\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]^{2}\mathds{1}(X<\mu)\Big{\\}}.$

There is $x_{0}>0$ such that $\log(1+x)\leq x^{1/2}$ for $x\geq x_{0}$. We
have

$\displaystyle\mathbb{E}\Big{\\{}[\log(1+\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]^{2}\mathds{1}(X\geq\mu)\Big{\\}}$
$\displaystyle\leq\mathbb{E}\Big{\\{}[\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2}/2]^{2}\mathds{1}(\mu\leq
X\leq\mu+x_{0}/\lambda_{i}\Big{\\}}$
$\displaystyle+\mathbb{E}\Big{\\{}[\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2}/2]^{2}\mathds{1}(X>\mu+x_{0}/\lambda_{i})\Big{\\}}$
$\displaystyle=\lambda_{i}^{2}\mathbb{E}[(X-\mu)^{2}\mathds{1}(\mu\leq
X\leq\mu+x_{0}/\lambda_{i})]+o(\lambda_{i}^{2})$
$\displaystyle=\lambda_{i}^{2}\mathbb{E}[(X-\mu)^{2}\mathds{1}(X\geq\mu)]+o(\lambda_{i}^{2}).$
(10)

Similarly,

$\displaystyle\mathbb{E}\Big{\\{}[\log(1-\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]^{2}\mathds{1}(X<\mu)\Big{\\}}=\lambda_{i}^{2}\mathbb{E}[(X-\mu)^{2}\mathds{1}(X<\mu)]+o(\lambda_{i}^{2}).$
(11)

From (6) and (11), we have that
$\mathbb{E}Y_{i}^{2}\leq\lambda_{i}^{2}\sigma^{2}+o(\lambda_{i}^{2})$. On the
other hand,

$\displaystyle\mathbb{E}Y_{i}^{2}$
$\displaystyle\geq\mathbb{E}\Big{\\{}[\log(1-\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]^{2}\mathds{1}(\mu\leq
X\leq\mu+x_{0}/\lambda_{i})\Big{\\}}$
$\displaystyle+\mathbb{E}\Big{\\{}[\log(1+\lambda_{i}(X-\mu)+\lambda_{i}^{2}(X-\mu)^{2})/2]^{2}\mathds{1}(\mu-
x_{0}/\lambda_{i}\leq X<\mu)\Big{\\}}.$

Similarly,
$\mathbb{E}Y_{i}^{2}\geq\lambda_{i}^{2}\sigma^{2}+o(\lambda_{i}^{2})$.
Therefore,

$\mathbb{E}Y_{i}^{2}=\lambda_{i}^{2}\sigma^{2}+o(\lambda_{i}^{2}).$

It follows from the above arguments that

$s_{n}^{2}\sim\sigma^{2}\sum_{i=1}^{n}\lambda_{i}^{2},~{}\text{and}~{}\theta_{n}:=s_{n}(2\log\log
s_{n}^{2})^{1/2}~{}\sim~{}\sigma\Big{(}2\sum_{i=1}^{n}\lambda_{i}^{2}\log\log\sum_{i=1}^{n}\lambda_{i}^{2}\Big{)}^{1/2}.$

We verify that the condition $(2,\alpha)$ in [16] holds for sequence
$\theta_{n}$ and $\vartheta$. Denoting by $c$ a generic positive constant that
may change from time to time, we have for large $n_{0}$,

$\sum_{n=n_{0}}^{\infty}\theta_{n}^{-(2+\vartheta)}\mathbb{E}|Y_{i}-\mathbb{E}Y_{i}|^{2+\vartheta}\leq
c\sum_{n=n_{0}}^{\infty}\Big{(}\sum_{i=1}^{n}\lambda_{i}^{2}\log\log\sum_{i=1}^{n}\lambda_{i}^{2}\Big{)}^{-1-\vartheta/2}\mathbb{E}|Y_{i}|^{2+\vartheta}.$

We have using $\log(1+x+x^{2}/2)\leq 2\log(1+x/\sqrt{2})$ and
$\log(1-x+x^{2}/2)\geq 2\log(1-x/\sqrt{2})$,

$\displaystyle\mathbb{E}|Y_{i}|^{2+\vartheta}$
$\displaystyle\leq\mathbb{E}[[2\log(1+\lambda_{i}(X-\mu)/\sqrt{2})]^{2+\vartheta}\mathds{1}(X>\mu)]$
$\displaystyle+\mathbb{E}[[2\log(1-\lambda_{i}(X-\mu)/\sqrt{2})]^{2+\vartheta}\mathds{1}(X<\mu)]\leq
c\lambda_{i}^{2+\vartheta}.$

Therefore, as ($\lambda_{n}$) is non-increasing,

$\sum_{n=n_{0}}^{\infty}\theta_{n}^{-(2+\vartheta)}\mathbb{E}|Y_{i}-\mathbb{E}Y_{i}|^{2+\vartheta}\leq
c\sum_{n=n_{0}}^{\infty}\Big{(}\sum_{i=1}^{n}\lambda_{i}^{2}\Big{)}^{-1-\vartheta/2}\lambda_{i}^{2+\vartheta}\leq
c\sum_{n=n_{0}}^{\infty}\frac{1}{n^{1+\vartheta/2}}<\infty.$

Hence the condition $(2,\alpha)$ in [16] holds. It follows that

$\limsup_{n\rightarrow\infty}\frac{s_{n+1}}{s_{n}}=\limsup_{n\rightarrow\infty}\Big{(}\frac{\sum_{i=1}^{n+1}\lambda_{i}^{2}}{\sum_{i=1}^{n}\lambda_{i}^{2}}\Big{)}^{1/2}\leq\limsup_{n\rightarrow\infty}\Big{(}\frac{n+1}{n}\Big{)}^{1/2}=1.$

Therefore, by [16, Theorem 2.1]

$\limsup_{n\rightarrow\infty}\theta_{n}^{-1}\sum_{i=1}^{n}(Y_{i}-\mathbb{E}Y_{i})=1,~{}\text{a.s.}$

That is,

$\limsup_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}(Y_{i}-\mathbb{E}Y_{i})}{\sigma\Big{(}2\sum_{i=1}^{n}\lambda_{i}^{2}\log\log\sum_{i=1}^{n}\lambda_{i}^{2}\Big{)}^{1/2}}=1,~{}\text{a.s.}$

This means that the sequence $(\widetilde{a}_{n})$ and $(\widetilde{b}_{n})$
must satisfy

$\widetilde{b}_{n}-\widetilde{a}_{n}>a\Big{(}2\sum_{i=1}^{n}\lambda_{i}^{2}\log\log\sum_{i=1}^{n}\lambda_{i}^{2}\Big{)}^{1/2}$

for any $a<2\sigma\sqrt{2}$ and large $n$, which implies the same for
$b_{n}-a_{n}$ and the result holds.

Proof of Theorem 3: For a fixed $x\in\mathbb{R}$ the following processes are
also non-negative supermartingales:

$\displaystyle
M_{n}^{+}(x)=\prod_{i=1}^{n}\exp\bigl{\\{}\phi\bigl{(}\lambda_{i}(X_{i}-x)\bigr{)}\bigr{\\}}$
(12) $\displaystyle\hskip
50.58878pt\exp\left\\{-(\mu-x)\sum_{i=1}^{n}\lambda_{i}-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}-C_{p}|\mu-x|^{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right\\}$

and

$\displaystyle
M_{n}^{-}(x)=\prod_{i=1}^{n}\exp\bigl{\\{}-\phi\bigl{(}\lambda_{i}(X_{i}-x)\bigr{)}\bigr{\\}}$
(13) $\displaystyle\hskip
50.58878pt\exp\left\\{(\mu-x)\sum_{i=1}^{n}\lambda_{i}-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}-C_{p}|\mu-x|^{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right\\}.$

Note that for $x=\mu$ and $t_{i}\equiv 1$, these processes become $M_{n}^{+}$
and $M_{n}^{-}$, correspondingly. Denote
$f_{n}(x)=\sum_{i=1}^{n}\phi\bigl{(}\lambda_{i}(X_{i}-x)\bigr{)}.$ The maximal
inequality for non-negative supermartingales, for every $x\in\mathbb{R}$ and
$h>0$,

$\displaystyle
P\left(\exp\left\\{f_{n}(x)-(\mu-x)\sum_{i=1}^{n}\lambda_{i}-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}-C_{p}|\mu-x|^{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right\\}\geq
h\right)$ $\displaystyle\leq$ $\displaystyle 1/h,$

which is the same as

$\displaystyle P\biggl{(}$ $\displaystyle
f_{n}(x)\geq(\mu-x)\sum_{i=1}^{n}\lambda_{i}+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}+C_{p}|\mu-x|^{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}+\log
h\biggr{)}\leq 1/h.$ (14)

Choose $h=2/\varepsilon_{n}$ for $0<\varepsilon_{n}<1$ and denote

$\displaystyle
B_{n}^{+}(x)=(\mu-x)\sum_{i=1}^{n}\lambda_{i}+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}+C_{p}|m-x|^{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}+\log
2/\varepsilon_{n}.$ (15)

Then (14) translates into

$\displaystyle P\Bigl{(}f_{n}(x)\geq
B_{n}^{+}(x)\Bigr{)}\leq\varepsilon_{n}/2.$ (16)

Consider now the equation

$B_{n}^{+}(x)=-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}-\log 2/\alpha.$ (17)

We will establish conditions under which this equation has real roots.
Assuming, for a moment, that such roots exist, let $y_{n}$ denote the smallest
such root. Using (17) with $x=y_{n}$ tells us that on an event of probability
at least $1-\varepsilon_{n}/2$, we have
$f_{n}(y_{n})<-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}-\log 2/\alpha.$ We
conclude by the definition of $x_{-,n}$ in (7), that

$P\Bigl{(}x_{-,n}<y_{n}\ \ \text{for all $n$ for which \eqref{e:eq.plus} has
real roots}\Bigr{)}\geq 1-\sum_{n=1}^{\infty}\varepsilon_{n}/2.$ (18)

We now establish conditions for the equation (17) to have real roots. The
function $B_{n}^{+}$ is a strictly convex function of $x$, diverging to
infinity at $\pm\infty$, so it has a unique minimum, achieved at the point

$z_{+}=\mu+\left(\frac{\sum_{i=1}^{n}\lambda_{i}}{pC_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}}\right)^{1/(p-1)},$

and we have

$B_{n}^{+}(z_{n})=-\frac{p}{p-1}\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(pC_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right)^{1/(p-1)}}+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}+\log
2/\varepsilon_{n}.$ (19)

If this minimal value satisfies

$\displaystyle-\frac{p}{p-1}\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(pC_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right)^{1/(p-1)}}+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}+\log
2/\varepsilon_{n}$ (20) $\displaystyle\leq$ $\displaystyle-
C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}-\log 2/\alpha,$

then the equation (17) has real roots. Note that we can rewrite the condition
(20) in the form

$\displaystyle C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1+t_{i}^{-(p-1)})+\log
2/\alpha+\log
2/\varepsilon_{n}\leq\frac{p}{p-1}\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(pC_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right)^{1/(p-1)}}.$
(21)

We claim that this condition holds for all large $n$, at least if $(t_{n})$
are bounded away from 0, and if $\varepsilon_{n}$ is not too small. Indeed, in
this case for some constant $C$,

$\sum_{i=1}^{n}\lambda_{i}^{p}(1+t_{i}^{-(p-1)})\leq
C\sum_{i=1}^{n}\lambda_{i}^{p},$ (22)

while

$\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right)^{1/(p-1)}}\geq\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(\sum_{i=1}^{n}\lambda_{i}^{p}\right)^{1/(p-1)}}.$
(23)

Since the ratio of the expressions in the right-hand sides of (23) and (22) is

$\frac{1}{C}\left(\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)}{\sum_{i=1}^{n}\lambda_{i}^{p}}\right)^{1/(p-1)}\to\infty$

by (6), we conclude that (21) holds and, hence, the equation (17) has real
roots, at least for all large $n$, as long $\varepsilon_{n}$ does not go to
zero too fast.

Notice that, if $\varepsilon_{n}\leq 2$, then

$B_{n}^{+}(\mu)=C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}t_{i}^{-(p-1)}+\log
2/\varepsilon_{n}>0>-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}-\log 2/\alpha.$

Furthermore, the minimum of $B_{n}^{+}$ is achieved to the right of $\mu$.
Therefore, under the condition (21), the equation (17) has one or two real
roots to the right of $\mu$, and $y_{n}$ is the smallest of these roots.

For $x>\mu$ the equation (17) becomes

$\displaystyle
C_{p}(x-\mu)^{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}-(x-\mu)\sum_{i=1}^{n}\lambda_{i}$
(24) $\displaystyle\hskip
108.405pt+C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\varepsilon_{n}+\log 2/\alpha=0.$

We can rewrite (24) in the form

$Kz^{p}-z+M=0$ (25)

for $z=x-\mu>0$ and

$K=\frac{C_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}}{\sum_{i=1}^{n}\lambda_{i}}$

and

$M=\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\varepsilon_{n}+\log 2/\alpha}{\sum_{i=1}^{n}\lambda_{i}}.$

Setting $y=K^{1/(p-1)}z>0$ and $D=K^{1/(p-1)}M$ transforms (25) into the
equation

$y^{p}-y+D=0.$ (26)

Let $\tau_{n}>0$ and suppose that

$D\leq\frac{\tau_{n}^{1/(p-1)}}{(1+\tau_{n})^{p/(p-1)}}.$ (27)

Then the equation (26) has a positive solution $y(D)$ satisfying

$y(D)\leq(1+\tau_{n})D,$

which implies that

$y_{n}\leq\mu+(1+\tau_{n})M=\mu+(1+\tau_{n})\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\varepsilon_{n}+\log 2/\alpha}{\sum_{i=1}^{n}\lambda_{i}}.$ (28)

Note that the condition (27) can be rewritten in the form

$\displaystyle C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1+t_{i}^{-(p-1)})+\log
2/\alpha+\log
2/\varepsilon_{n}\leq\frac{\tau_{n}^{1/(p-1)}}{(1+\tau_{n})^{p/(p-1)}}\frac{\left(\sum_{i=1}^{n}\lambda_{i}\right)^{p/(p-1)}}{\left(C_{p}\sum_{i=1}^{n}\lambda_{i}^{p}(1-t_{i})^{-(p-1)}\right)^{1/(p-1)}}.$
(29)

Similarly to the condition (21), this condition holds for all large $n$ as
long as $\varepsilon_{n}$ and $\tau_{n}$ do not go to zero too fast. We
conclude by (18) and (28)

$\displaystyle
P\biggl{(}x_{-,n}<\mu+(1+\tau_{n})\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\varepsilon_{n}+\log 2/\alpha}{\sum_{i=1}^{n}\lambda_{i}}$ (30)
$\displaystyle\hskip 108.405pt\text{for all $n$ for which \eqref{e:condD.sol}
holds}\biggr{)}\geq 1-\sum_{i=1}^{\infty}\varepsilon_{i}/2.$

The same argument shows that

$\displaystyle
P\biggl{(}x_{+,n}>\mu-(1+\tau_{n})\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\varepsilon_{n}+\log 2/\alpha}{\sum_{i=1}^{n}\lambda_{i}}$ (31)
$\displaystyle\hskip 108.405pt\text{for all $n$ for which \eqref{e:condD.sol}
holds}\biggr{)}\geq 1-\sum_{i=1}^{\infty}\varepsilon_{i}/2.$

We conclude by (30) and (31) that

$\displaystyle P\biggl{(}\bigl{|}I_{n}(\alpha)\bigr{|}\leq
2(1+\tau_{n})\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\varepsilon_{n}+\log 2/\alpha}{\sum_{i=1}^{n}\lambda_{i}}$ (32)
$\displaystyle\hskip 108.405pt\text{for all $n$ for which \eqref{e:condD.sol}
holds}\biggr{)}\geq 1-\sum_{n=1}^{\infty}\varepsilon_{n}.$

With $\varepsilon_{n}$ as in the statement, (32) is transformed into

$\displaystyle P\biggl{(}\bigl{|}I_{n}(\alpha)\bigr{|}\leq
4(1+\tau_{n})\frac{C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}+\log
2/\alpha}{\sum_{i=1}^{n}\lambda_{i}}$ (33) $\displaystyle\hskip
50.58878pt\text{for all $n$ for which \eqref{e:condD.sol} holds}\biggr{)}\geq
1-\alpha\sum_{n=1}^{\infty}\exp\left\\{-C_{p}v_{p}\sum_{i=1}^{n}\lambda_{i}^{p}\bigl{(}1+t_{i}^{-(p-1)}\bigr{)}\right\\}.$

It follows from (6) that the sum in the right hand side is finite, and can be
made small if $\alpha$ is small.

##  7 Conclusion

We provided an extension of confidence sequences for settings where the
variance of the data-generating distribution does not exist. Dealing with such
heavy-tail settings required using robust estimation methods to obtain
acceptable deviation bounds. We made use of the influence functions inspired
by [1] to obtain Catoni-style confidence sequences. We first established lower
bounds on the widths of the Catoni-style confidence sequences for the finite
variance case using a general law of iterated logarithm. We then provided a
simple confidence sequence using a $L_{p}-$version of the Dubins-Savage
inequality for comparison. Finally, we derived Catoni-style confidence
sequences in case of infinite variance and strengthened the existing result on
finite variance in [14].

## References

  * [1] Olivier Catoni.  Challenging the empirical mean and empirical variance: a deviation study.  In Annales de l’IHP Probabilités et statistiques, volume 48, pages 1148–1185, 2012.
  * [2] Peng Chen, Xinghu Jin, Xiang Li, and Lihu Xu.  A generalized catoni’s m-estimator under finite $\alpha$-th moment assumption with $\alpha\in$(1, 2).  Electronic Journal of Statistics, 15(2):5523–5544, 2021.
  * [3] Donald A Darling and Herbert Robbins.  Confidence sequences for mean, variance, and median.  Proceedings of the National Academy of Sciences of the United States of America, 58(1):66, 1967.
  * [4] Lester E Dubins and Leonard J Savage.  A tchebycheff-like inequality for stochastic processes.  Proceedings of the National Academy of Sciences of the United States of America, 53(2):274, 1965.
  * [5] Steven R Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon.  Time-uniform, nonparametric, nonasymptotic confidence sequences.  The Annals of Statistics, 49(2):1055–1080, 2021.
  * [6] Kevin G Jamieson and Lalit Jain.  A bandit approach to sequential experimental design with false discovery control.  Advances in Neural Information Processing Systems, 31, 2018.
  * [7] Christopher Jennison and Bruce W Turnbull.  Interim analyses: the repeated confidence interval approach.  Journal of the Royal Statistical Society: Series B (Methodological), 51(3):305–334, 1989.
  * [8] Ramesh Johari, Pete Koomen, Leonid Pekelis, and David Walsh.  Peeking at a/b tests: Why it matters, and what to do about it.  In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1517–1525, 2017.
  * [9] Ramesh Johari, Leo Pekelis, and David J Walsh.  Always valid inference: Bringing sequential analysis to a/b testing.  arXiv preprint arXiv:1512.04922, 2015.
  * [10] Olav Kallenberg.  On the existence and path properties of stochastic integrals.  The Annals of Probability, pages 262–280, 1975.
  * [11] Rasul A Khan.  $l_{p}$-version of the dubins–savage inequality and some exponential inequalities.  Journal of Theoretical Probability, 22(2):348–364, 2009.
  * [12] Alan Malek and Silvia Chiappa.  Asymptotically best causal effect identification with multi-armed bandits.  Advances in Neural Information Processing Systems, 34, 2021.
  * [13] Herbert Robbins.  Some aspects of the sequential design of experiments.  Bulletin of the American Mathematical Society, 58(5):527–535, 1952\.
  * [14] Hongjian Wang and Aaditya Ramdas.  Catoni-style confidence sequences for heavy-tailed mean estimation.  arXiv preprint arXiv:2202.01250, 2022.
  * [15] Ian Waudby-Smith and Aaditya Ramdas.  Estimating means of bounded random variables by betting.  arXiv preprint arXiv:2010.09686, 2020.
  * [16] Rainer Wittmann.  A general law of iterated logarithm.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 68(4):521–543, 1985.
  * [17] Ruohan Zhan, Vitor Hadad, David A Hirshberg, and Susan Athey.  Off-policy evaluation via adaptive weighting with data from contextual bandits.  In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 2125–2135, 2021.


# On the continuity of the solution map of the Euler-Poincaré equations in
Besov spaces

Min Li1 111E-mail<EMAIL_ADDRESS>
1 Department of Mathematics, Jiangxi University of Finance and Economics,
Nanchang, 330032, China

Abstract: By constructing a series of perturbation functions through
localization in the Fourier domain and using a symmetric form of the system,
we show that the data-to-solution map for the Euler-Poincaré equations is
nowhere uniformly continuous in $B^{s}_{p,r}(\mathbb{R}^{d})$ with
$s>\max\\{1+\frac{d}{2},\frac{3}{2}\\}$ and
$(p,r)\in(1,\infty)\times[1,\infty)$. This improves our previous result which
shows the data-to-solution map for the Euler-Poincaré equations is non-
uniformly continuous on a bounded subset of $B^{s}_{p,r}(\mathbb{R}^{d})$ near
the origin.

Keywords: Euler-Poincaré equations; Nowhere uniformly continuous; Besov
spaces; Data-to-solution map.

MSC (2010): 35Q35, 35Q51, 35L30

##  1 Introduction

In this paper, we consider the Cauchy problem in $\mathbb{R}^{d}$ for Euler-
Poincaré equations

$\begin{cases}\partial_{t}m+u\cdot\nabla m+\nabla u^{T}\cdot
m+(\mathrm{div}u)m=0,\qquad&(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d},\\\
m=(1-\Delta)u,\qquad&(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d},\\\
u(0,x)=u_{0},\qquad&x\in\mathbb{R}^{d}.\end{cases}$ (1.1)

The equations (1.1) were first introduced by Holm, Marsden, and Ratiu in [17,
18] as a high dimensional generalization of the following Camassa-Holm
equation for modeling and analyzing the nonlinear shallow water waves :

$\displaystyle m_{t}+um_{x}+2u_{x}m=0,\ m=u-u_{xx}.$ (CH)

Indeed, when $d=1$ the Euler-Poincaré equations are the same as the Camassa-
Holm equation (CH). Also, the Euler-Poincaré equations were investigated as
the system describe geodesic motion on the diffeomorphism group with respect
to the kinetic energy norm in [16].

For $d=1$, the equation (CH) was introduced by Camassa and Holm[5] as a bi-
Hamiltonian model for shallow water waves. Most importantly, CH equation has
peakon solutions of the form $Ce^{-|x-Ct|}$ which aroused a lot of interest in
physics, see [8, 29]. There is an extensive literature about the strong well-
posedness, weak solutions and analytic or geometric properties of the CH
equation, here we name some. Local well-posedness and ill-posedness for the
Cauchy problem of the CH equation were investigated in [9, 12, 13]. Blow-up
phenomena and global existence of strong solutions were discussed in [7, 9,
10, 11]. The existence of global weak solutions and dissipative solutions were
investigated in [3, 4, 30], more results can be found in the references
therein.

The first rigorous analysis of the Euler-Poincaré equations (1.1) was done by
Chae and Liu [6], they eatablished the local existence of weak solution in
$W^{2,p}(\mathbb{R}^{d}),\ p>d$ and local existence of unique classical
solutions in $H^{s}(\mathbb{R}^{d}),\ s>\frac{d}{2}+3$. Yan and Yin [31]
further discussed the local existence and uniqueness of the solution to (1.1)
in Besov spaces. On the other hand, Li, Yu and Zhai [27] proved that the
solutions to (1.1) with a large class of smooth initial data blows up in
finite time or exists globally in time, which settled an open problem raised
by Chae and Liu [6]. Later, Luo and Yin have obtained a new blow-up result in
the periodic case by using the rotational invariant properties of the
equation[25]. For more results of Euler-Poincaré equations, see [25, 32].

Recently, starting from the research of Himonas et al. [14, 15], the
continuity properties of the data-to-solution maps of the Camassa-Holm type
equations are gradually attracting interest of many authors, see [21, 23].
Most of the non-uniform constinuity results are established only on a bounded
set near the origin. To overcome this limitation, Inci obtained a series of
nowhere uniform continuity results including many Camassa-Holm type equations
[19, 20]. And for the incompressible Euler equation, Bourgain and Li [2]
showed that the data-to-solution map is nowhere-uniform continuity in
$H^{s}(\mathbb{R}^{d})$ with $s\geq 0$ by using an idea of localized Galilean
boost, this method will inspire us in this article.

As part of the well-posedness theory, the continuity properties of the data-
to-solution map is indeed very important. In fact, the non-uniform continuity
of data-to-solution map suggests that the local well-posedness cannot be
established by the contraction mappings principle since this would imply
Lipschitz continuity for the solution map. On the other hand, in some critical
spaces the continuity of the data-to-solution maps are first broken before the
existence and uniqueness of the solution, which leads to ill-posedness [22].

Most previous work on constinuity has focused on the spacial one-dimensional
Camassa-Holm type equations equations, for the multi-dimensional Euler-
Poincaré equations (1.1), the continuity problem has not been thoroughly
investigated. Until recently, Li et al. [24] shown that the corresponding
solution to (1.1) is not uniformly constinuous dependence for that the initial
data in $H^{s}(\mathbb{R}^{d}),s>1+\frac{d}{2}$. Later, the non-uniformly
constinuous result was extended to Besov space
$B^{s}_{p,r}(\mathbb{R}^{d}),s>\max\\{1+\frac{d}{2},\frac{3}{2}\\}$ in [26].

It is worth to mention that, the non-uniform constinuity results of (1.1) are
established only on a bounded set near the origin. In this paper, we will
remove the boundedness restriction and prove that the data-to-solution map of
the Euler-Poincaré equations (1.1) is not uniformly continuous on any open
subset $U\subset
B^{s}_{p,r}(\mathbb{R}^{d}),s>\max\\{1+\frac{d}{2},\frac{3}{2}\\}$.
Technically, our proof based on a symmetric form of the equation (1.1), and a
translation method to construct perturbation data, this method was introduced
by Bourgain and Li [2] to proof the nowhere uniform constinuity of the
incompressible Euler equations.

For simplicity, we first transform Eq.(1.1) into a transport type system.
According to Yan[31], we can rewrite (1.1) to the following nonlocal form:

$\displaystyle\partial_{t}u+u\cdot\nabla u=Q(u,u)+R(u,u),$ (1.2)

where

$\begin{cases}&Q(u,v)=-(I-\Delta)^{-1}\mathrm{div}\Big{(}\nabla u\nabla
v+\nabla u\nabla v^{T}-\nabla u^{T}\nabla v-\nabla
u(\mathrm{div}v)+\frac{1}{2}(\nabla u:\nabla v)\mathbf{I}\Big{)},\\\
&R(u,v)=-(I-\Delta)^{-1}\big{(}u~{}\mathord{{\rm div}}v+\nabla
u^{T}v\big{)}.\end{cases}$ (1.3)

We now define a symmetric bilinear operator $\mathcal{T}$ by

$\displaystyle\mathcal{T}(u,v)$
$\displaystyle:=\frac{1}{2}\Big{(}Q(u,v)+Q(v,u)+R(u,v)+R(v,u)\Big{)}$
$\displaystyle=-(I-\Delta)^{-1}\mathrm{div}\big{(}M(\nabla u,\nabla
v)\big{)}-(I-\Delta)^{-1}\big{(}N(u,\nabla u;v,\nabla v)\big{)},$ (1.4)

here $M,N$ are bilinear functions of $(\nabla u,\nabla v)$ and $(u,\nabla
u;v,\nabla v)$ respectively according to (1.3), they are symmetric on $u,v$.
Then, the Euler-Poincaré equations becomes

$\begin{cases}\partial_{t}u+u\cdot\nabla
u=\mathcal{T}(u,u),\qquad&(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d},\\\
u(0,x)=u_{0},\qquad&x\in\mathbb{R}^{d}.\end{cases}$ (E-P)

We first recall the non-uniform continuity results established in [26].

######  Theorem 1.1 (Non-uniform continuity on a bounded set).

Let $d\geq 2$ and $s>2+\max\big{\\{}1+\frac{d}{p},\frac{3}{2}\big{\\}}$ with
$1\leq p,r\leq\infty$. The data-to-solution map $S_{t}$ for Euler-Poincaré
equations (E-P) is not uniformly continuous from any bounded subset
$O_{N}=\\{u_{0}\in B^{s}_{p,r}(\mathbb{R}^{d}):\|u_{0}\|_{B^{s}_{p,r}}\leq
N\\}$ into $\mathcal{C}([0,T];B^{s}_{p,r})$. More precisely, there exists two
sequences of initial data $f_{n}+g_{n},~{}f_{n}$ such that

$\displaystyle\|f_{n}\|_{B^{s}_{p,r}}\lesssim 1\quad
and\quad\lim_{n\to\infty}\|g_{n}\|_{B^{s}_{p,r}}=0,$

with the solutions $S_{t}(f_{n}+g_{n}),~{}S_{t}(f_{n})$ satisfy

$\displaystyle\liminf_{n\to\infty}\|S_{t}(f_{n}+g_{n})-S_{t}(f_{n})\|_{B^{s}_{p,r}}\geq
c_{0}t,~{}\forall t\in[0,T_{0}],$

for some constant $c_{0}>0$ and small time $T_{0}$.

The main result of this paper is the following theorem.

######  Theorem 1.2 (Nowhere uniform continuity).

Assume that $d\geq 2$, and

$\displaystyle s>2+\max\big{\\{}1+\frac{d}{p},\frac{3}{2}\big{\\}}\quad
and\quad(p,r)\in(1,\infty)\times[1,\infty).$ (1.5)

Then the data-to-solution map $S_{t}$ for Euler-Poincaré equations for the
Cauchy problem (E-P)

$\displaystyle
S_{t}:B^{s}_{p,r}(\mathbb{R}^{d})\to\mathcal{C}([0,T];B^{s}_{p,r}),\quad
u_{0}\mapsto S_{t}(u_{0}),$

is nowhere uniformly continuous from $B^{s}_{p,r}$ into
$\mathcal{C}([0,T];B^{s}_{p,r})$. More precisely, for any $u_{0}\in
B^{s}_{p,r}$ and $N>0$, there exists two sequences of functions
$f_{n}(x),g_{n}(x)$ such that

$\displaystyle\|f_{n}\|_{B^{s}_{p,r}}\lesssim 2^{-N}\quad
and\quad\lim_{n\to\infty}\|g_{n}\|_{B^{s}_{p,r}}=0,$

the corresponding solutions $S_{t}(f_{n}+g_{n}),~{}S_{t}(f_{n})$ satisfy

$\displaystyle\liminf_{n\to\infty}\|S_{t}(u_{0}+f_{n}+g_{n})-S_{t}(u_{0}+f_{n})\|_{B^{s}_{p,r}}\geq
c_{0}t,~{}\forall t\in[0,T_{0}],$

for some constant $c_{0}>0$ and small time $T_{0}$.

######  Remark 1.1.

As a comparison with Theorem 1.1, Theorem 1.2 avoids endpoints $p=1$ and
$p=\infty$, this is because we need to use the boundedness of Riez transform
in $L^{p}(\mathbb{R}^{d})$ when doing gradient estimate of $\mathcal{T}$ (see
Lemma 3.2 blow), which is only available when $p\in(1,\infty)$.

######  Remark 1.2.

The non-uniform constinuity in Theorem 1.1 established only on a bounded set
near the origin, in Theorem 1.2 we have removed these restrictions and showed
that for any $u_{0}$ and any neighbour $U(u_{0})\subset B^{s}_{p,r}$, the
data-to-solution map restrict on $U$ is not uniformly continuous. In this
sense, Theorem 1.2 improves the previous results in [26].

The remainder of this paper is organized as follows. In Section 2, we list
some notations and recall basic results of the Littlewood-Paley theory. In
Section 3, we present the proof of Theorem 1.2 by establishing some technical
lemmas and propositions.

##  2 Littlewood-Paley analysis

We first present some facts about the Littlewood-Paley decomposition, the
nonhomogeneous Besov spaces and their some useful properties (see [1] for more
details).

Let $\mathcal{B}:=\\{\xi\in\mathbb{R}^{d}:|\xi|\leq 4/3\\}$ and
$\mathcal{C}:=\\{\xi\in\mathbb{R}^{d}:3/4\leq|\xi|\leq 8/3\\}.$ Choose a
radial, non-negative, smooth function $\chi:\mathbb{R}^{d}\mapsto[0,1]$ such
that it is supported in $\mathcal{B}$ and $\chi\equiv 1$ for $|\xi|\leq 3/4$.
Setting $\varphi(\xi):=\chi(\xi/2)-\chi(\xi)$, then we deduce that $\varphi$
is supported in $\mathcal{C}$. Moreover,

$\displaystyle\chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1\quad\mbox{ for any
}\xi\in\mathbb{R}^{d}.$

We should emphasize that the fact $\varphi(\xi)\equiv 1$ for $4/3\leq|\xi|\leq
3/2$ will be used in the sequel.

For every $u\in\mathcal{S^{\prime}}(\mathbb{R}^{d})$, the inhomogeneous dyadic
blocks ${\Delta}_{j}$ are defined as follows

${\Delta_{j}u=}\begin{cases}0,&if\quad j\leq-2;\\\
\chi(D)u=\mathcal{F}^{-1}(\chi\mathcal{F}u),&if\quad j=-1;\\\
\varphi(2^{-j}D)u=\mathcal{F}^{-1}\big{(}\varphi(2^{-j}\cdot)\mathcal{F}u\big{)},&if\quad
j\geq 0.\end{cases}$

In the inhomogeneous case, the following Littlewood-Paley decomposition makes
sense

$u=\sum_{j\geq-1}{\Delta}_{j}u\quad\text{for
any}\;u\in\mathcal{S^{\prime}}(\mathbb{R}^{d}).$

######  Definition 2.1.

labelbesov Let $s\in\mathbb{R}$ and $(p,r)\in[1,\infty]^{2}$. The
nonhomogeneous Besov space $B^{s}_{p,r}(\mathbb{R}^{d})$ is defined by

$\displaystyle
B^{s}_{p,r}(\mathbb{R}^{d}):=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}^{d}):\;\|f\|_{B^{s}_{p,r}(\mathbb{R}^{d})}<\infty\Big{\\}},$

where

$\displaystyle\|f\|_{B^{s}_{p,r}(\mathbb{R}^{d})}=$
$\displaystyle\left(\sum_{j\geq-1}2^{sjr}\|\Delta_{j}f\|^{r}_{L^{p}(\mathbb{R}^{d})}\right)^{\frac{1}{r}},$
if $1\leq r<\infty$, $\displaystyle\|f\|_{B^{s}_{p,r}(\mathbb{R}^{d})}=$
$\displaystyle\sup_{j\geq-1}2^{sj}\|\Delta_{j}f\|_{L^{p}(\mathbb{R}^{d})},$ if
$r=\infty$.

The following Bernstein’s inequalities will be used in the sequel.

######  Lemma 2.1.

Let $\mathcal{B}$ be a Ball and $\mathcal{C}$ be an annulus. There exist
constants $C>0$ such that for all $k\in\mathbb{N}\cup\\{0\\}$, any positive
real number $\lambda$ and any function $f\in L^{p}(\mathbb{R}^{d})$ with
$1\leq p\leq q\leq\infty$, we have

$\displaystyle{\rm{supp}}\hat{f}\subset\lambda\mathcal{B}\;\Rightarrow\;\|D^{k}f\|_{L^{q}}:=\sup_{|\alpha|=k}\|\partial^{\alpha}f\|_{L^{q}}\leq
C^{k+1}\lambda^{k+(\frac{d}{p}-\frac{d}{q})}\|f\|_{L^{p}},$
$\displaystyle{\rm{supp}}\hat{f}\subset\lambda\mathcal{C}\;\Rightarrow\;C^{-k-1}\lambda^{k}\|f\|_{L^{p}}\leq\|\Delta^{k}f\|_{L^{p}}\leq
C^{k+1}\lambda^{k}\|f\|_{L^{p}}.$

######  Lemma 2.2 (See [1]).

Let $(s_{1},s_{2},p,r)\in\mathbb{R}^{2}\times[1,\infty]^{2}$, and
$s_{1}<s_{2},~{}0<\theta<1$, then we have

$\displaystyle\|u\|_{B^{\theta s_{1}+(1-\theta)s_{2}}_{p,r}}\leq$
$\displaystyle\|u\|_{B^{s_{1}}_{p,r}}^{\theta}\|u\|_{B^{s_{2}}_{p,r}}^{1-\theta},$
$\displaystyle\|u\|_{B^{\theta s_{1}+(1-\theta)s_{2}}_{p,1}}\leq$
$\displaystyle\frac{C}{s_{2}-s_{1}}\Bigl{(}\frac{1}{\theta}+\frac{1}{1-\theta}\Bigr{)}\|u\|_{B^{s_{1}}_{p,\infty}}^{\theta}\|u\|_{B^{s_{2}}_{p,\infty}}^{1-\theta}.$

Then, we give some important product estimates which will be used throughout
the paper.

######  Lemma 2.3 (See [1]).

For $(p,r)\in[1,\infty]^{2}$ and $s>0$, $B^{s}_{p,r}(\mathbb{R}^{d})\cap
L^{\infty}(\mathbb{R}^{d})$ is an algebra. Moreover, for any $u,v\in
B^{s}_{p,r}(\mathbb{R}^{d})\cap L^{\infty}(\mathbb{R}^{d})$, we have

$\displaystyle\|uv\|_{B^{s}_{p,r}}\leq
C(\|u\|_{B^{s}_{p,r}}\|v\|_{L^{\infty}}+\|v\|_{B^{s}_{p,r}}\|u\|_{L^{\infty}}).$

In addition, if $s>\max\big{\\{}1+\frac{d}{p},\frac{3}{2}\big{\\}}$, then

$\displaystyle\|uv\|_{B^{s-2}_{p,r}(\mathbb{R}^{d})}\leq
C\|u\|_{B^{s-2}_{p,r}(\mathbb{R}^{d})}\|v\|_{B^{s-1}_{p,r}(\mathbb{R}^{d})}.$

######  Lemma 2.4 (See [1, 28]).

Let $(p,r)\in[1,\infty]^{2}$ and
$\sigma\geq-\min\big{\\{}\frac{d}{p},1-\frac{d}{p}\big{\\}}$. Assume that
$f_{0}\in B^{\sigma}_{p,r}(\mathbb{R}^{d})$, $g\in
L^{1}([0,T];B^{\sigma}_{p,r}(\mathbb{R}^{d}))$ and $\nabla\mathbf{u}\in
L^{1}([0,T];B^{\sigma-1}_{p,r}(\mathbb{R}^{d}))$ if $\sigma>1+\frac{d}{p}$ or
$\sigma=1+\frac{d}{p},r=1$. If $f\in
L^{\infty}([0,T];B^{\sigma}_{p,r}(\mathbb{R}^{d}))\cap\mathcal{C}([0,T];\mathcal{S}^{\prime}(\mathbb{R}^{d}))$
solves the following linear transport equation:

$\quad\partial_{t}f+\mathbf{u}\cdot\nabla f=g,\quad\;f|_{t=0}=f_{0}.$

1\. There exists a constant $C=C(\sigma,p,r)$ such that the following
statement holds

$\displaystyle\|f(t)\|_{B^{\sigma}_{p,r}}\leq
e^{CV(t)}\Big{(}\|f_{0}\|_{B^{\sigma}_{p,r}}+\int_{0}^{t}e^{-CV(\tau)}\|g(\tau)\|_{B^{\sigma}_{p,r}}\mathrm{d}\tau\Big{)},$

where

$V(t)=\int_{0}^{t}\|\nabla\mathbf{u}(\tau)\|_{B^{\sigma-1}_{p,r}}\mathrm{d}\tau\quad\text{if}\quad\sigma>1+{\frac{d}{p}}\quad\text{or}\quad\\{\sigma=1+\frac{d}{p},\;r=1\\}.$

2\. If $\sigma>0$, then there exists a constant $C=C(\sigma,p,r)$ such that
the following holds

$\displaystyle\|f(t)\|_{B^{\sigma}_{p,r}}\leq$
$\displaystyle\|f_{0}\|_{B^{\sigma}_{p,r}}+\int_{0}^{t}\|g(\tau)\|_{B^{\sigma}_{p,r}}\mathrm{d}\tau$
$\displaystyle+\int_{0}^{t}\Big{(}\|f(\tau)\|_{B^{\sigma}_{p,r}}\|\nabla\mathbf{u}\|_{L^{\infty}}+\|\nabla\mathbf{u}\|_{B^{\sigma-1}_{p,r}}\|\nabla
f(\tau)\|_{L^{\infty}}\Big{)}\mathrm{d}\tau.$

##  3 Proof of the main theorem

We first recall the local existence and uniqueness theory of solutions for the
Cauchy problem (1.1) in Besov spaces [31], then provide some technical lemmas
and propositions.

###  3.1 Preparation and technical lemmas

######  Lemma 3.1 (See [31]).

Assume that

$\displaystyle d\in\mathbb{N}_{+},1\leq
p,r\leq\infty~{}and~{}s>\max\\{1+\frac{d}{p},\frac{3}{2}\\}.$ (3.1)

Let $u_{0}\in B^{s}_{p,r}(\mathbb{R}^{d})$, then there exists a time
$T=T(\|u_{0}\|_{B^{s}_{p,r}(\mathbb{R}^{d})})>0$ such that (1.1) has a unique
solution in

$\begin{cases}C([0,T];B^{s}_{p,r}(\mathbb{R}^{d}))\cap
C^{1}([0,T];B^{s-1}_{p,r}(\mathbb{R}^{d})),&if~{}~{}r<\infty,\\\
L^{\infty}([0,T];B^{s}_{p,\infty}(\mathbb{R}^{d}))\cap
Lip([0,T];B^{s-1}_{p,\infty}(\mathbb{R}^{d})),&if~{}~{}r=\infty.\end{cases}$

And the mapping $u_{0}\mapsto u$ is continuous from
$B^{s}_{p,r}(\mathbb{R}^{d})$ into
$C([0,T];B^{s^{\prime}}_{p,r}(\mathbb{R}^{d}))\cap
C^{1}([0,T];B^{s^{\prime}-1}_{p,r}(\mathbb{R}^{d}))$ for all $s^{\prime}<s$ if
$r=\infty$, and $s^{\prime}=s$ otherwise. Moreover, for all $t\in[0,T]$, there
holds

$\displaystyle\|u(t)\|_{B^{s}_{p,r}(\mathbb{R}^{d})}\leq
C\|u_{0}\|_{B^{s}_{p,r}(\mathbb{R}^{d})}.$

######  Lemma 3.2.

Let $(s,p,r)$ satisfy (1.5), then for the symmetric bilinear operator
$\mathcal{T}(f,g)$ defined by (1.3) and (1.4), we have

$\displaystyle\|\mathcal{T}(f,g)\|_{B^{s}_{p,r}}\leq
C\|f\|_{B^{s}_{p,r}}\|g\|_{B^{s}_{p,r}}$ (3.2)

If $0<p<\infty$, there holds

$\displaystyle\|\mathcal{T}(f,g)\|_{L^{p}}$ $\displaystyle\leq\|\nabla
f\|_{L^{p}}\|g,\nabla g\|_{L^{\infty}}$ (3.3)
$\displaystyle\|\mathcal{T}(f,g)\|_{L^{p}}$
$\displaystyle\leq\sum_{0\leq|a|,|b|\leq
1}\|\partial^{a}f\partial^{b}g\|_{L^{p}}=W_{1,p}(f,g)$ (3.4)

And, for the gradient $\nabla\mathcal{T}$, we have

$\displaystyle\|\nabla\mathcal{T}(f,g)\|_{L^{p}}$ $\displaystyle\leq\|\nabla
f\|_{L^{p}}\|g,\nabla g\|_{L^{\infty}}$ (3.5)
$\displaystyle\|\nabla\mathcal{T}(f,g)\|_{L^{p}}$
$\displaystyle\leq\sum_{0\leq|a|,|b|\leq
2}\|\partial^{a}f\partial^{b}g\|_{L^{p}}=W_{2,p}(f,g)$ (3.6)

Where we denote $\displaystyle{W_{m,p}(f,g)=\sum_{0\leq|a|,|b|\leq
m}\|\partial^{a}f\partial^{b}g\|_{L^{p}}}$ with the multiindex
$a=(a_{1},a_{2},\cdots,a_{d}),~{}|a|=a_{1}+\cdots+a_{d}$ and
$\partial^{a}=\frac{\partial^{|a|}}{\partial x_{1}^{a_{1}}\cdots\partial
x_{d}^{a_{d}}}$.

###### Proof.

As the operator $(I-\Delta)^{-1}$ is a Fourier $S^{-2}$-multiplier, it’s easy
to see that

$\displaystyle\|\mathcal{T}(f,g)\|_{B^{s}_{p,r}}\leq C\|M(\nabla f,\nabla
g)\|_{B^{s-1}_{p,r}}+C\|N(f,\nabla f,g,\nabla g)\|_{B^{s-2}_{p,r}}\leq
C\|f\|_{B^{s}_{p,r}}\|g\|_{B^{s}_{p,r}},$

here we have use the Lemma 2.3. Then in $L^{p}$ spaces,

$\displaystyle\|\mathcal{T}(f,g)\|_{L^{p}}$
$\displaystyle=\|(I-\Delta)^{-1}\mathrm{div}\big{(}M(\nabla f,\nabla
g)\big{)}+(I-\Delta)^{-1}\big{(}N(f,\nabla f,g,\nabla g)\big{)}\|_{L^{p}}$
$\displaystyle\leq\|M(\nabla f,\nabla g)\|_{L^{p}}+\|N(f,\nabla f,g,\nabla
g)\|_{L^{p}}$ $\displaystyle\leq\|\nabla f\|_{L^{p}}\|\nabla
g\|_{L^{\infty}}+\|\nabla f\|_{L^{p}}\|g\|_{L^{\infty}}$
$\displaystyle\leq\|\nabla f\|_{L^{p}}\|g,\nabla g\|_{L^{\infty}}$

we also have

$\displaystyle\|\mathcal{T}(f,g)\|_{L^{p}}$ $\displaystyle\leq\|M(\nabla
f,\nabla g)\|_{L^{p}}+\|N(f,\nabla f,g,\nabla g)\|_{L^{p}}$
$\displaystyle\leq\sum_{0\leq|a|,|b|\leq
1}\|\partial^{a}f\partial^{b}g\|_{L^{p}}=W_{1,p}(f,g)$

For the gradient $\nabla\mathcal{T}$, noting that
$(I-\Delta)^{-1}\partial_{i}\partial_{j}=-\Delta(I-\Delta)^{-1}\big{(}(-\Delta)^{-1}\partial_{i}\partial_{j}\big{)}=\big{(}(1-\Delta)^{-1}+1\big{)}R_{i}R_{j}$
and the Riesz transform $R_{i}$ is bounded in $L^{p}\to
L^{p},~{}p\in(1,\infty)$, then we have

$\displaystyle\|\nabla\mathcal{T}(f,g)\|_{L^{p}}$
$\displaystyle=\|\nabla(I-\Delta)^{-1}\mathrm{div}\big{(}M(\nabla f,\nabla
g)\big{)}+\nabla(I-\Delta)^{-1}\big{(}N(f,\nabla f,g,\nabla
g)\big{)}\|_{L^{p}}$ $\displaystyle\leq\|M(\nabla f,\nabla
g)\|_{L^{p}}+\|N(f,\nabla f,g,\nabla g)\|_{L^{p}}$ $\displaystyle\leq\|\nabla
f\|_{L^{p}}\|\nabla g\|_{L^{\infty}}+\|\nabla f\|_{L^{p}}\|g\|_{L^{\infty}}$
$\displaystyle\leq\|\nabla f\|_{L^{p}}\|g,\nabla g\|_{L^{\infty}}$

and

$\displaystyle\|\nabla\mathcal{T}(f,g)\|_{L^{p}}$
$\displaystyle\leq\|\mathord{{\rm div}}M(\nabla f,\nabla
g)\|_{L^{p}}+\|N(f,\nabla f,g,\nabla g)\|_{L^{p}}$
$\displaystyle\leq\sum_{0\leq|a|,|b|\leq
2}\|\partial^{a}f\partial^{b}g\|_{L^{p}}=W_{2,p}(f,g)$

∎

We’ll need the following estimates of the difference $u(t)-v(t)$ in Besov
spaces.

######  Proposition 3.1.

Let $1\leq p,r\leq\infty$ and $s>\max\\{1+\frac{d}{p},\frac{3}{2}\\}$. Assume
that $u(t),v(t)$ are solutions of (E-P) with initial data $(u_{0},v_{0})\in
B^{s}_{p,r}(\mathbb{R}^{d})$, then $\delta(t):=u(t)-v(t)$ satisfies

$\displaystyle\|\delta(t)\|_{B^{s-1}_{p,r}}\leq\|\delta_{0}\|_{B^{s-1}_{p,r}}\exp\big{(}C\int_{0}^{t}\|u(\tau),v(\tau)\|_{B^{s}_{p,r}}d\tau\big{)}$

and

$\displaystyle\|\delta(t)\|_{B^{s}_{p,r}}\leq\Big{(}\|\delta_{0}\|_{B^{s}_{p,r}}+C\int_{0}^{t}\|\delta\|_{B^{s-1}_{p,r}}\|\nabla
v\|_{B^{s}_{p,r}}d\tau\Big{)}\exp\big{(}C\int_{0}^{t}\|u(\tau),v(\tau)\|_{B^{s}_{p,r}}d\tau\big{)}$
(3.7)

###### Proof.

The first inequality has been proved in [31], it remains to prove (3.7). As
$\mathcal{T}$ is a symmetric bilinear operator, it’s easy to deduce that
$\delta=u-v$ solves the transport equation

$\displaystyle\partial_{t}\delta+u\cdot\nabla\delta=-\delta\cdot\nabla
v+\mathcal{T}(\delta,u+v).$ (3.8)

Then, by Lemma 2.4 and 3.2

$\displaystyle\|\delta(t)\|_{B^{s}_{p,r}}\leq$
$\displaystyle\|\delta_{0}\|_{B^{s}_{p,r}}+C\int_{0}^{t}\Big{(}\|u\|_{B^{s}_{p,r}}\|\delta\|_{B^{s}_{p,r}}+\|\delta\cdot\nabla
v\|_{B^{s}_{p,r}}+\|\mathcal{T}(\delta,u+v)\|_{B^{s}_{p,r}}\Big{)}d\tau$
$\displaystyle\leq$
$\displaystyle\|\delta_{0}\|_{B^{s}_{p,r}}+C\int_{0}^{t}\Big{(}\|u(\tau),v(\tau)\|_{B^{s}_{p,r}}\|\delta\|_{B^{s}_{p,r}}+\|\delta\|_{B^{s-1}_{p,r}}\|\nabla
v\|_{B^{s}_{p,r}}\Big{)}d\tau.$

now (3.7) is direct result from Gronwall’s inequality. ∎

######  Proposition 3.2.

Suppose $\widetilde{u}(t),u(t),v(t)$ are the solutions of (E-P) of initial
data $u_{0}+v_{0},u_{0},v_{0}$ respectively. Then, under the assumptions of
(1.5), we have

$\displaystyle\|\widetilde{u}-u-v\|_{B^{s}_{p,r}}\leq
C\|u_{0},v_{0}\|_{B^{s+1}_{p,\infty}}^{1-\theta}\exp\Big{(}\|u_{0},v_{0}\|_{B^{s+1}_{p,r}}\theta\Big{)}\Big{(}\int_{0}^{t}W_{2,p}(u,v)d\tau\Big{)}^{\theta},$

where $\theta=\frac{1}{s+1}$ and use the notation
$\displaystyle{W_{2,p}(u,v)=\sum_{0\leq|a|,|b|\leq
2}\|\partial^{a}u\partial^{b}v\|_{L^{p}}}$.

###### Proof.

Since $\widetilde{u}(t),u(t),v(t)$ are solutions of

$\begin{cases}\partial_{t}\widetilde{u}+\widetilde{u}\cdot\nabla\widetilde{u}=\mathcal{T}(\widetilde{u},\widetilde{u}),\qquad&\widetilde{u}(0)=u_{0}+v_{0},\\\
\partial_{t}u+u\cdot\nabla u=\mathcal{T}(u,u),\qquad&u(0)=u_{0},\\\
\partial_{t}v+v\cdot\nabla v=\mathcal{T}(v,v),\qquad&v(0)=v_{0}.\end{cases}$

by the symmetry and linearity of $\mathcal{T}$, we can deduce that
$w(t)=\widetilde{u}(t)-u(t)-v(t)$ satisfies

$\begin{cases}\partial_{t}w+\widetilde{u}\cdot\nabla
w=&-w\cdot\nabla(u+v)+\mathcal{T}(w,\widetilde{u}+u+v)\\\ &-u\cdot\nabla
v-v\cdot\nabla u-2\mathcal{T}(u,v),\\\ \qquad\qquad w(0)=&0.\end{cases}$ (3.9)

By the interpolation inequality (see Lemma 2.2 ), we obtain

$\displaystyle\|w\|_{B^{s}_{p,r}}\leq
C\|w\|_{B^{0}_{p,\infty}}^{\theta}\|w\|_{B^{s+1}_{p,\infty}}^{1-\theta}\leq\|u_{0},v_{0}\|_{B^{s+1}_{p,\infty}}^{1-\theta}\|w\|_{L^{p}}^{\theta}$
(3.10)

The rest of the proof is to bound the $L^{p}$ norm of $w$, taking the inner
product of (3.9) with
$\widetilde{w}^{p-1}:=(|w_{1}|^{p-2}w_{1},|w_{2}|^{p-2}w_{2},\cdots,|w_{d}|^{p-2}w_{d})$,
we obtain

$\displaystyle\frac{1}{p}\frac{d}{dt}\|w\|_{L^{p}}^{p}=$
$\displaystyle\sum_{i=1}^{d}\int p^{-1}|w_{i}|^{p}(\mathord{{\rm
div}}\widetilde{u})dx-\sum_{i,j}\int\widetilde{w}_{i}^{p-1}w_{j}\partial_{j}(u_{i}+v_{i})dx$
$\displaystyle+\sum_{i=1}^{d}\int\widetilde{w}_{i}^{p-1}\mathcal{T}_{i}(w,\widetilde{u}+u+v)dx-\sum_{i=1}^{d}\int\widetilde{w}_{i}^{p-1}(u\cdot\nabla
v_{i}+v\cdot\nabla u_{i}+\mathcal{T}_{i}(u,v)dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{p}\|\mathord{{\rm
div}}\widetilde{u}\|_{L^{\infty}}\|w\|_{L^{p}}^{p}+C_{d}(\|\nabla
u\|_{L^{\infty}}+\|\nabla v\|_{L^{\infty}})\|w\|_{L^{p}}^{p}$
$\displaystyle+\|w\|_{L^{p}}^{p-1}\|\mathcal{T}(w,\widetilde{u}+u+v)\|_{L^{p}}+\|w\|_{L^{p}}^{p-1}\|u\cdot\nabla
v+v\cdot\nabla u+\mathcal{T}(u,v)\|_{L^{p}}$ (3.11)

Thanks to the estimates of $\mathcal{T}$ in Lemma 3.2, in particular take
(3.3), (3.4), into (3.1) we have

$\displaystyle\frac{d}{dt}\|w\|_{L^{p}}$ $\displaystyle\leq C(\|\mathord{{\rm
div}}\widetilde{u}\|_{L^{\infty}}+\|\nabla u\|_{L^{\infty}}+\|\nabla
v\|_{L^{\infty}})\|w\|_{L^{p}}$
$\displaystyle+C(\|\widetilde{u},\nabla\widetilde{u}\|_{L^{\infty}}+\|u,\nabla
u\|_{L^{\infty}}+\|v,\nabla v\|_{L^{\infty}})\|\nabla
w\|_{L^{p}}+W_{2,p}(u,v)$
$\displaystyle\leq\|u_{0},v_{0}\|_{B^{s}_{p,r}}(\|w\|_{L^{p}}+\|\nabla
w\|_{L^{p}})+W_{2,p}(u,v)$

Now, we should bound the gradient matrix $\nabla w$, take the gradient to
(3.9), then in components

$\displaystyle\partial_{t}\partial_{j}w_{i}=$
$\displaystyle-\widetilde{u}_{k}\partial_{k}\partial_{j}w_{i}-\partial_{j}\widetilde{u}_{k}\partial_{k}w_{i}+\partial_{j}T_{i}(w,\widetilde{u}+u+v)$
$\displaystyle-
w_{k}\partial_{k}(\partial_{j}u_{i}+\partial_{j}v_{i})-\partial_{j}w_{k}\partial_{k}(u_{i}+v_{i})-\partial_{j}\big{(}u_{k}\partial_{k}v_{i}-v_{k}\partial_{k}u_{i}-2\mathcal{T}_{i}(u,v)\big{)}$

Taking the $L^{2}$ inner product with
$\widetilde{w}_{i,j}^{p-1}:=|\partial_{j}w_{i}|^{p-2}\partial_{j}w_{i}$ and
sum the indices $i,j$, we get

$\displaystyle\frac{1}{p}\frac{d}{dt}\|\nabla w\|_{L^{p}}^{p}=$
$\displaystyle\sum_{1\leq i,j\leq d}\int
p^{-1}|\partial_{j}w_{i}|^{p}(\mathord{{\rm
div}}\widetilde{u})dx-\int\nabla\widetilde{w}^{p-1}:(\nabla
w\nabla\widetilde{u})dx+\int\nabla\widetilde{w}^{p-1}:\nabla\mathcal{T}(w,\widetilde{u}+u+v)dx$
$\displaystyle-\int\nabla\widetilde{w}^{p-1}:\big{(}w\cdot\nabla(\nabla
u+\nabla v)\big{)}dx-\int\nabla\widetilde{w}^{p-1}:\big{(}(\nabla u+\nabla
v)\nabla w\big{)}dx$
$\displaystyle-\int\nabla\widetilde{w}^{p-1}:\nabla(u\cdot\nabla
v+v\cdot\nabla u+2\mathcal{T}(u,v))dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{p}\|\mathord{{\rm
div}}\widetilde{u}\|_{L^{\infty}}\|\nabla
w\|_{L^{p}}^{p}+C_{d}\|\nabla\widetilde{u}\|_{L^{\infty}}\|\nabla
w\|_{L^{p}}^{p}+\|\nabla
w\|_{L^{p}}^{p-1}\|\nabla\mathcal{T}(w,\widetilde{u}+u+v)\|_{L^{p}}$
$\displaystyle+\|\nabla
w\|_{L^{p}}^{p-1}\|w\|_{L^{p}}\big{(}\|\nabla^{2}u\|_{L^{\infty}}+\|\nabla^{2}v\|_{L^{\infty}}\big{)}+\|\nabla
w\|_{L^{p}}^{p}\big{(}\|\nabla u\|_{L^{\infty}}+\|\nabla
v\|_{L^{\infty}}\big{)}$ $\displaystyle+|\nabla
w\|_{L^{p}}^{p-1}\|\nabla\big{(}u\cdot\nabla v+v\cdot\nabla
u+2\mathcal{T}(u,v)\big{)}\|_{L^{p}}$ (3.12)

where we denote
$\nabla\widetilde{w}^{p-1}=(\widetilde{w}_{i,j}^{p-1})_{d\times d}$ and
$A:B:=\sum_{i,j}a_{i,j}b_{i,j}$. Again using Proposition 3.2 for the matrix
operator $\nabla\mathcal{T}$, by plug (3.5),(3.6) into (3.1), we obtain

$\displaystyle\frac{d}{dt}\|\nabla w\|_{L^{p}}$ $\displaystyle\leq
C(\|\nabla\widetilde{u}\|_{L^{\infty}}+\|\nabla u\|_{L^{\infty}}+\|\nabla
v\|_{L^{\infty}})\|\nabla
w\|_{L^{p}}+\|w\|_{L^{p}}\big{(}\|\nabla^{2}u\|_{L^{\infty}}+\|\nabla^{2}v\|_{L^{\infty}}\big{)}$
$\displaystyle+\|\nabla\mathcal{T}(w,\widetilde{u}+u+v)\|_{L^{p}}+\|\nabla\big{(}u\cdot\nabla
v+v\cdot\nabla u+2\mathcal{T}(u,v)\big{)}\|_{L^{p}}$ $\displaystyle\leq
C(\|\widetilde{u},\nabla\widetilde{u}\|_{L^{\infty}}+\|u,\nabla
u\|_{L^{\infty}}+\|v,\nabla v\|_{L^{\infty}})\|\nabla w\|_{L^{p}}$
$\displaystyle+\big{(}\|\nabla^{2}u\|_{L^{\infty}}+\|\nabla^{2}v\|_{L^{\infty}}\big{)}\|w\|_{L^{p}}+W_{2,p}(u,v)$
$\displaystyle\leq C\|u_{0},v_{0}\|_{B^{s+1}_{p,r}}(\|w\|_{L^{p}}+\|\nabla
w\|_{L^{p}})+W_{2,p}(u,v)$

Combining (3.14) and (3.17) yields that

$\displaystyle\frac{d}{dt}\|w,\nabla w\|_{L^{p}}$ $\displaystyle\leq
C\|u_{0},v_{0}\|_{B^{s+1}_{p,r}}(\|w\|_{L^{p}}+\|\nabla
w\|_{L^{p}})+2W_{2,p}(u,v)$

By Gronwall’s inequality and (3.10) we complete the proof. ∎

######  Remark 3.1.

The proofs of Proposition 3.1 and 3.2 rely on the symmetry of $\mathcal{T}$,
especially when it comes to getting simplified equations (3.8) and (3.9). Most
previous studies on the well-posedness of Euler-Poincaré equations use the
bilinear form (1.2), the lack of symmetry makes the calculation complicated.
Infact, when $d=1$ namely the Camassa-Holm equation has the transport form
$\partial_{t}u+u\partial_{x}u=P(u,u)$ with
$P(u,v)=-\partial_{x}(1-\partial_{x}^{2})^{-1}\bigl{(}uv+\frac{1}{2}(\partial_{x}u\partial_{x}v)\bigr{)}$
is symmetric by default. In this respect, our new form (E-P) is a more natural
high-dimensional generalization of the CH equation.

###  3.2 Construction of Perturbation Data

For localization in the Fourier domain, we introduce the following bump
function in the frequency space. Let
$\widehat{\phi}\in\mathcal{C}^{\infty}_{0}(\mathbb{R})$ be a non-negative and
even function satisfy

$\displaystyle\widehat{\phi}(\xi)=$ $\displaystyle 1,$ if
$|\xi|\leq\frac{1}{4}$,  $\displaystyle\widehat{\phi}(\xi)=$ $\displaystyle
0,$ if $|\xi|\geq\frac{1}{2}$.

and let

$\begin{cases}&f_{n}=2^{-ns-N}\big{(}\cos(\frac{17}{12}2^{n}x_{1})\phi(x_{1})\phi(x_{2})\cdots\phi(x_{d}),0,\cdots,0\big{)}\\\
&g_{n}=\big{(}2^{-n}\phi(x_{1})\phi(x_{2})\cdots\phi(x_{d}),0,\cdots,0\big{)}.\end{cases}$
(3.13)

We define the perturbation data by adding a translation transform

$\begin{cases}&f_{n}^{m}=f_{n}(x_{1}-m,x_{2},\cdots,x_{d})\\\
&g_{n}^{m}=g_{n}(x_{1}-m,x_{2},\cdots,x_{d})\end{cases}$ (3.14)

Noting that $\widehat{f_{n}^{m}}$ is supported in
$[-\frac{1}{2},\frac{1}{2}]^{d}\pm(\frac{17}{12}2^{n},0,\cdots,0),$ this
support set is completely covered by the ring
$C_{n}=\\{\xi\in\mathbb{R}^{d}:\frac{4}{3}2^{n}\leq|\xi|\leq\frac{3}{2}2^{n}\\}.$
Thus, by the definition of $\Delta_{j},$ we know

$\displaystyle\Delta_{j}(f_{n})=$ $\displaystyle f_{n}^{m},$ if $j=n$,
$\displaystyle\Delta_{j}(f_{n})=$ $\displaystyle 0,$ if $j\neq n$. (3.15)

On account of above and the definition of Besov space, we can show that for
$k\in\mathbb{R}$

$\displaystyle\|f_{n}^{m}\|_{B^{s+k}_{p,r}}\leq C2^{kn-N}\qquad
and\qquad\|g_{n}^{m}\|_{B^{s+k}_{p,r}}\to 0\qquad for~{}n\rightarrow\infty$
(3.16)

By the previous work [26] and translation invariance of the system (E-P), we
know that, for the corresponding solutions $S_{t}(f_{n}^{m}+g_{n}^{m})$ and
$S_{t}(f_{n}^{m})$ there is a positive constant $c_{0}$ and a small time
$T_{0}$, such that for any $t\in[0,T_{0}]$,

$\displaystyle\liminf_{n\to\infty}\|S_{t}(f_{n}^{m}+g_{n}^{m})-S_{t}(f_{n}^{m})\|_{B^{s}_{p,r}}\geq
c_{0}t.$ (3.17)

###  3.3 Proof of Theorem 1.2

Roughly speaking, our proof of Theorem 1.2 based on the following
approximation

$\displaystyle\qquad\qquad S_{t}(u_{0}+f_{n}^{m}+g_{n}^{m})$ $\displaystyle-
S_{t}(u_{0}+f_{n}^{m})$ (I)
$\displaystyle=S_{t}(S_{n}u_{0}+f_{n}^{m}+g_{n}^{m})$ $\displaystyle-
S_{t}(S_{n}u_{0}+f_{n}^{m})+\mathcal{E}_{n}^{m}$ (II)
$\displaystyle=\Big{(}S_{t}(S_{n}u_{0})+S_{t}(f_{n}^{m}+g_{n}^{m})\Big{)}$
$\displaystyle-\Big{(}S_{t}(S_{n}u_{0})+S_{t}(f_{n}^{m})\Big{)}+\mathcal{E}_{n,m}$
$\displaystyle=S_{t}(f_{n}^{m}+g_{n}^{m})$ $\displaystyle-
S_{t}(f_{n}^{m})+\mathcal{E}_{n,m},$ (III)

with some small error terms $\mathcal{E}_{n}^{m},~{}\mathcal{E}_{n,m}$. More
precisely, we devide (I) into three parts

$\displaystyle\begin{split}&S_{t}(u_{0}+f_{n}^{m}+g_{n}^{m})-S_{t}(u_{0}+f_{n}^{m})=\\\
&\underbrace{\Big{(}S_{t}(u_{0}+f_{n}^{m}+g_{n}^{m})-S_{t}(S_{n}u_{0}+f_{n}^{m}+g_{n}^{m})\Big{)}-\Big{(}S_{t}(u_{0}+f_{n}^{m})-S_{t}(S_{n}u_{0}+f_{n}^{m})\Big{)}}_{\mathcal{E}_{n}^{m}}+\\\
&\underbrace{\Big{(}S_{t}(S_{n}u_{0}+f_{n}^{m}+g_{n}^{m})-S_{t}(S_{n}u_{0})-S_{t}(f_{n}^{m}+g_{n}^{m})\Big{)}-\Big{(}S_{t}(S_{n}u_{0}+f_{n}^{m})-S_{t}(S_{n}u_{0})-S_{t}(f_{n}^{m})\Big{)}}_{\mathcal{E}_{n,m}}\\\
&+S_{t}(f_{n}^{m}+g_{n}^{m})-S_{t}(f_{n}^{m}).\end{split}$ (3.18)

We proof the approximation $(\ref{C})\to(\ref{B})\to(\ref{A})$ in the
following sense.

######  Proposition 3.3.

Let $f_{n}^{m},~{}g_{n}^{m}$ be the perturbation data defined by (3.13) and
(3.14), then for any initial data $u_{0}\in B^{s}_{p,r}$ with
$\|u_{0}\|_{B^{s}_{p,r}}=\rho$, the error terms
$\mathcal{E}_{n}^{m},~{}\mathcal{E}_{n,m}$ in (3.18) satisfy

$\displaystyle\sup_{m,t}\|\mathcal{E}_{n}^{m}\|_{B^{s}_{p,r}}\leq
C_{\rho}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}},$ (3.19)
$\displaystyle\lim_{m\to\infty}\bigl{(}\sup_{0\leq t\leq
T}\|\mathcal{E}_{n,m}\|_{B^{s}_{p,r}}\bigr{)}=0\quad for~{}any~{}fixed~{}n.$
(3.20)

.

###### Proof.

We first to handle (3.19). Using proposition 3.1 with
$\delta(t)=S_{t}(u_{0}+f_{n}^{m})-S_{t}(S_{n}u_{0}+f_{n}^{m})$, as
$\|u_{0}+f_{n}^{m}\|_{B^{s}_{p,r}}\approx\|S_{n}u_{0}+f_{n}^{m}\|_{B^{s}_{p,r}}$
for $m\in\mathbb{R}$ and $n\gg 1$, the solution sequences have a common
lifespan $T\approx T^{*}(\|u_{0}\|_{B^{s}_{p,r}})$, then for any $t\in[0,T)$
we have

$\displaystyle\|\delta(t)\|_{B^{s}_{p,r}}\leq$
$\displaystyle\Big{(}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}+\int_{0}^{t}\|\delta(\tau)\|_{B^{s-1}_{p,r}}\|\nabla
S_{t}(S_{n}u_{0}+f_{n}^{m})\|_{B^{s}_{p,r}}d\tau\Big{)}$
$\displaystyle\cdot\exp\big{(}\int_{0}^{t}\|S_{t}(u_{0}+f_{n}^{m}),S_{t}(S_{n}u_{0}+f_{n}^{m})\|_{B^{s}_{p,r}}d\tau\big{)}$
$\displaystyle\leq$
$\displaystyle\Big{(}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}+\int_{0}^{t}\|\delta(\tau)\|_{B^{s-1}_{p,r}}\|S_{n}u_{0}+f_{n}^{m}\|_{B^{s+1}_{p,r}}d\tau\Big{)}$
$\displaystyle\cdot\exp\big{(}\int_{0}^{t}\|u_{0}+f_{n}^{m},S_{n}u_{0}+f_{n}^{m}\|_{B^{s}_{p,r}}d\tau\big{)}$
$\displaystyle\leq$ $\displaystyle
C_{\rho}\Big{(}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}+\int_{0}^{t}\|\delta(\tau)\|_{B^{s-1}_{p,r}}\cdot
2^{n}d\tau\Big{)}$ (3.21)

and

$\displaystyle\|\delta(t)\|_{B^{s-1}_{p,r}}\leq$
$\displaystyle\|(I-S_{n})u_{0}\|_{B^{s-1}_{p,r}}\exp\big{(}\int_{0}^{t}\|S_{t}(u_{0}+f_{n}^{m}),S_{t}(S_{n}u_{0}+f_{n}^{m})\|_{B^{s}_{p,r}}d\tau\big{)}$
$\displaystyle\leq$ $\displaystyle
C_{\rho}2^{-n}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}$ (3.22)

take (3.3) into (3.3) we get

$\displaystyle\|\delta(t)\|_{B^{s}_{p,r}}\leq
C_{\rho}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}.$ (3.23)

As in (3.23) the $C_{\rho}$ not depend on the translation parameter $m$ and
$t\in[0,T]$, then we have

$\displaystyle\sup_{m,t}\|S_{t}(u_{0}+f_{n}^{m})-S_{t}(S_{n}u_{0}+f_{n}^{m})\|_{B^{s}_{p,r}}=\sup_{m,t}\|\delta(t)\|_{B^{s}_{p,r}}\leq
C_{\rho}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}.$ (3.24)

With exactly the same argument, we can deduce that

$\sup_{m,t}\|S_{t}(u_{0}+f_{n}^{m}+g_{n}^{m})-S_{t}(S_{n}u_{0}+f_{n}^{m}+g_{n}^{m})\|_{B^{s}_{p,r}}\leq
C_{\rho}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}},$

along with (3.24), we complete the proof of (3.19).

In order to deduce (3.20), we should use Proposition 3.2 with the setting
$\widetilde{u}(t)=S_{t}(S_{n}u_{0}+f_{n}^{m}),u(t)=S_{t}(S_{n}u_{0})$ and
$v(t)=S_{t}(f_{n}^{m}),$ and denote

$w=\widetilde{u}-u-v=S_{t}(S_{n}u_{0}+f_{n}^{m})-S_{t}(S_{n}u_{0})-S_{t}(f_{n}^{m}).$

Since $u_{0}\in B^{s}_{p,r}$ and $\|f_{n}^{m}\|_{B^{s}_{p,r}}\approx 1$, it’s
easy to see that

$\displaystyle\|S_{n}u_{0},f_{n}^{m}\|_{B^{s+1}_{p,r}}\leq C_{\rho}2^{n},$
(3.25)

with $C_{\rho}$ only depend on $\rho:=\|u_{0}\|_{B^{s}_{p,r}}$, Then from
Proposition 3.2 we know that

$\displaystyle\|w(t)\|_{B^{s}_{p,r}}\leq
C_{\rho}2^{n}e^{C_{\rho}2^{n}\theta}\Bigl{(}\sum_{0\leq|a|,|b|\leq
2}\int_{0}^{t}\|\partial^{a}S_{t}(S_{n}u_{0})\partial^{b}S_{t}(f_{n}^{m})\|_{L^{p}}d\tau\Bigr{)}^{\theta}.$
(3.26)

Notice that, by definition
$\partial^{b}S_{t}(f_{n}^{m})=\partial^{b}S_{t}(f_{n}(x_{1}-m,\cdots,x_{d}))=\partial^{b}S_{t}(f_{n})(x_{1}-m,\cdots,x_{d})$,
for fixed $n$ and any $(t,x)$, considering that $S_{t}(f_{n})$ is a smooth
function decay at infinity, we have

$\displaystyle\lim_{m\to\infty}\partial^{a}S_{t}(S_{n}u_{0})(x)\partial^{b}S_{t}(f_{n})(x_{1}-m,\cdots,x_{d})=0,$
$\displaystyle|\partial^{a}S_{t}(S_{n}u_{0})(x)\partial^{b}S_{t}(f_{n})(x_{1}-m,\cdots,x_{d})|\leq
M|\partial^{a}S_{t}(S_{n}u_{0})|(\tau,x)\in
L^{1}\bigl{(}[0,T],L^{p}(\mathbb{R})\bigr{)}.$

By the Lebesgue Dominated Convergence Theorem, we have

$\displaystyle\lim_{m\to\infty}\int_{0}^{T}\|\partial^{a}S_{t}(S_{n}u_{0})\partial^{b}S_{t}(f_{n}^{m})\|_{L^{p}}d\tau=0$
(3.27)

Then from (3.26),(3.27) we know that, for the fixed $n$ and any $t\in[0,T]$

$\displaystyle\lim_{m\to\infty}\sup_{0\leq t\leq
T}\|w(t)\|_{B^{s}_{p,r}}=\lim_{m\to\infty}\sup_{0\leq t\leq
T}\|S_{t}(S_{n}u_{0}+f_{n}^{m})-S_{t}(S_{n}u_{0})-S_{t}(f_{n}^{m})\|_{B^{s}_{p,r}}=0,$
(3.28)

with the same argument, we can also get that, for any fixed $n$ and
$t\in[0,T]$

$\displaystyle\lim_{m\to\infty}\sup_{0\leq t\leq
T}\|S_{t}(S_{n}u_{0}+f_{n}^{m}+g_{n}^{m})-S_{t}(S_{n}u_{0})-S_{t}(f_{n}^{m}+g_{n}^{m})\|_{B^{s}_{p,r}}=0.$
(3.29)

Combining (3.28) and (3.29), this yields (3.20). ∎

With (3.17), (3.18) and Propositions 3.3 in hand, we can complete our proof of
Theorem 1.2. First of all, from the identity (3.18) we know that for any time
$t\in[0,T]$

$\displaystyle\begin{split}&\quad\|S_{t}(u_{0}+f_{n}^{m}+g_{n}^{m})-S_{t}(u_{0}+f_{n}^{m})\|_{B^{s}_{p,r}}\\\
&\geq\|S_{t}(f_{n}^{m}+g_{n}^{m})-S_{t}(f_{n}^{m})\|_{B^{s}_{p,r}}-\sup_{m,t}\|\mathcal{E}_{n}^{m}\|_{B^{s}_{p,r}}-\sup_{0\leq
t\leq T}\|\mathcal{E}_{n,m}\|_{B^{s}_{p,r}}\end{split}$ (3.30)

Then, by (3.20) in Proposition 3.3, for the fixed $n$, we can find a
sufficiently large $m_{n}$ such that

$\displaystyle\sup_{0\leq t\leq T}\|\mathcal{E}_{n,m_{n}}\|_{B^{s}_{p,r}}\leq
2^{-n}.$

combining this and (3.19) in Proposition 3.3, by (3.30) we get

$\displaystyle\quad\|S_{t}(u_{0}+f_{n}^{m_{n}}+g_{n}^{m_{n}})-S_{t}(u_{0}+f_{n}^{m_{n}})\|_{B^{s}_{p,r}}$
$\displaystyle\geq\|S_{t}(f_{n}^{m_{n}}+g_{n}^{m_{n}})-S_{t}(f_{n}^{m_{n}})\|_{B^{s}_{p,r}}-C_{\rho}\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}-2^{-n}.$
(3.31)

As $u_{0}\in B^{s}_{p,r}$, that means $\|(I-S_{n})u_{0}\|_{B^{s}_{p,r}}\to 0$
when $n\to\infty$. For the small $t\in[0,T_{0}]$, we already have (3.17), it
follows from (3.3) that

$\displaystyle\liminf_{n\to\infty}\|S_{t}(u_{0}+f_{n}^{m_{n}}+g_{n}^{m_{n}})-S_{t}(u_{0}+f_{n}^{m_{n}})\|_{B^{s}_{p,r}}\geq
c_{0}t,\quad\forall t\in[0,T_{0}].$ (3.32)

And the sequences of initial data satisfy

$\displaystyle\lim_{n\to\infty}\|(u_{0}+f_{n}^{m_{n}}+g_{n}^{m_{n}})-(u_{0}+f_{n}^{m_{n}})\|_{B^{s}_{p,r}}=\lim_{n\to\infty}\|g_{n}^{m_{n}}\|_{B^{s}_{p,r}}=0.$
(3.33)

This complete the proof Theorem 1.2.

Acknowledgements. M. Li was supported by Educational Commission Science
Programm of Jiangxi Province (No. GJJ190284) and Natural Science Foundation of
Jiangxi Province (No. 20212BAB211011 and 20212BAB201008).

## References

  * [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011.
  * [2] J. Bourgain and D. Li, Galilean Boost and Non-uniform Continuity for Incompressible Euler, Commun. Math. Phys., 372 (2019), 261–280.
  * [3] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
  * [4] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
  * [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
  * [6] D. Chae and J. Liu, Blow-up, zero $\alpha$ limit and the Liouville type theorem for the Euler-Poincaré equations, Comm. Math. Phys., 314 (2012), 671-687.
  * [7] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
  * [8] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
  * [9] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
  * [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
  * [11] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328.
  * [12] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.
  * [13] Z. Guo, X. Liu, M. Luc and Z. Yin, Ill-posedness of the Camassa-Holm and related equations in the critical space, J. Differential Equations, 266 (2019), 1698-1707.
  * [14] A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Integral Eqns, 22 (2009), 201–224.
  * [15] A. Himonas, C. Kenig and Misiołek Non-uniform dependence for the periodic CH equation, Commun. Partial Diff. Eqns, 35 (2010), 1145–1162.
  * [16] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323–380.
  * [17] D. D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 80 (2007), 4173–4177.
  * [18] D. D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1–81.
  * [19] H. Inci, On the well-posedness of the Holm-Staley b-family of equations, Journal of Nonlinear Mathematical Physics, 23 (2016), 213–233.
  * [20] H. Inci, On the local well-posedness of the two component b-family of equations, Monatsh. Math., 197 (2022), 479–492.
  * [21] J. Li, Y. Yu and W. Zhu, Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces, J. Differ. Equ., 269 (2020), 8686–8700.
  * [22] J. Li, Y. Yu and W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differ. Equ., 306 (2022), 403–417.
  * [23] J. Li, X. Wu, Y. Yu and W. Zhu , Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space, J. Math. Fluid Mech., 23 (2021), 1422-6928.
  * [24] J. Li, L. Dai and W. Zhu, Non-uniform continuous dependence on initial data of solutions to the Euler-Poincaré system, J. Math. Phys., 60 (2019), 111510, 9.
  * [25] W. Luo and Z. Yin, Blow-up phenomena, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations, J. of Differential Equations, 268 (2020), 1307–1325.
  * [26] J. Li, W. Deng and M. Li , Non-uniform dependence for Euler-Poincaré equations in Besov spaces, Nonlinear Analysis: Real World Applications, 63 (2022), 103420.
  * [27] D. Li, X. Yu and Z. Zhai, On the Euler-Poincare equation with non-zero dispersion, Arch. Ration. Mech. Anal., 210 (2013), 955-974.
  * [28] J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.
  * [29] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
  * [30] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
  * [31] K. Yan and Z. Yin, On the initial value problem for Euler-Poincaré equations, Discrete Contin. Dyn. Syst., 35 (2015), 1327-1358.
  * [32] Y. Zhao, M. Yang and Y. Li, Non-uniform dependence for the periodic Euler-Poincaré equations, J. Math. Anal. Appl., 461 (2018), 59-73.