# Pseudo-independence, independence and related limit theorems under sublinear
expectations

Li Xinpeng  Research Center for Mathematics and Interdisciplinary Sciences,
Shandong University, 266237, Qingdao, China; School of Mathematics, Shandong
University, 250100, Jinan, China<EMAIL_ADDRESS>
###### Abstract.

This paper introduces the notion of pseudo-independence on the sublinear
expectation space $(\Omega,\mathcal{F},\mathcal{P})$ via the classical
conditional expectation, and the relations between pseudo-independence and
Peng’s independence are detailed discussed. Law of large numbers and central
limit theorem for pseudo-independent random variables are obtained, and a
purely probabilistic proof of Peng’s law of large numbers is also given. In
the end, some relevant counterexamples are indicated that Peng’s law of large
numbers and central limit theorem are invalid only with the first and second
moment conditions respectively.

###### 2020 Mathematics Subject Classification:

Primary 60A05; Secondary 60F05

This work was supported by National Key R&D Program of China
(No.2018YFA0703900), NSF of China (No.11601281) and the Young Scholars Program
of Shandong University. The author gratefully acknowledges the many helpful
suggestions of Prof. Shige PENG during the preparation of the paper.

##  1\. Introduction

The notion of independence is very important in the classical probability
theory, especially in the limit theorems, which was builded on the probability
space $(\Omega,\mathcal{F},P)$, where the classical probability $P$ is
additive. However, in some areas this apparently natural property has been
abandoned. Non-additive probabilities or nonlinear expectations as the useful
tools for studying uncertainties have received more and more attention in many
fields, such as mathematical economics, statistics, quantum mechanics and
finance. One typical example of nonlinear expectations is the sublinear
expectation, Peng [10] systematically studied the sublinear expectation theory
and related stochastic analysis. In particular, the new notions of
independence and identical distribution (i.i.d. for short) are introduced in
Peng [10] and the related law of large numbers (LLN for short) and central
limit theorem (CLT for short) under sublinear expectation are obtained. These
notions and theorems are builded on the sublinear expectation space
$(\Omega,\mathcal{H},\mathbb{E})$, where $\mathcal{H}$ is a linear space of
random variables satisfying certain “closure property” for a class of
continuous functions and $\mathbb{E}$ is a sublinear functional on
$\mathcal{H}$.

In most situations, the sublinear expectation can be represented as the
supremum of a set $\mathcal{P}$ of linear expectations, i.e.,
$\mathbb{E}^{\mathcal{P}}[X]=\sup_{P\in\mathcal{P}}E_{P}[X]$, see Denis et al.
[2] or Peng [10]. The sublinear expectation space of this paper is slightly
different from the original one in Peng [10]. We consider
$(\Omega,\mathcal{F},\mathcal{P})$ as the sublinear expectation space, where
$\mathcal{P}$ is a set of probability measures on the measurable space
$(\Omega,\mathcal{F})$. Following Peng’s idea for the notion of independence
under sublinear expectation space $(\Omega,\mathcal{H},\mathbb{E})$, we
reconsider this notion on sublinear expectation space
$(\Omega,\mathcal{F},\mathcal{P})$, in particular, the set $\mathcal{P}$ can
be constructed such that Peng’s independence also holds under the sublinear
expectation $\mathbb{E}^{\mathcal{P}}$ introduced by $\mathcal{P}$. We first
introduce the notion of pseudo-independence under sublinear expectation
$\mathbb{E}^{\mathcal{P}}$ via conditional expectations. Then we can enlarge
$\mathcal{P}$ to $\bar{\mathcal{P}}$ such that pseudo-independent sequence
becomes the independent one under $\bar{\mathcal{P}}$. Thanks to the existing
results of LLN and CLT, we can easily obtain the corresponding LLN and CLT for
pseudo-independent sequence. In particular, a purely probabilistic proof of
Peng’s LLN for i.i.d. sequence is given. But unlike the classical limit
theorems, LLN and CLT for i.i.d. sequence may fail only with the finite first
and second moment conditions respectively.

The remainder of this paper is organized as follows. Section 2 describes some
basic concepts and notations of the sublinear expectation theory. In Section
3, the properties of pseudo-independent sequence are discussed. LLN and CLT
for pseudo-independent random variables are proved in Section 4 and Section 5.
The relevant counterexamples of LLN and CLT are provided in Section 6.

##  2\. Preliminaries

Given a complete separable metric space $\Omega$, let $\mathcal{F}$ be the
Borel $\sigma$-algebra of $\Omega$ and $\mathcal{M}$ be the collection of all
probability measures on $(\Omega,\mathcal{F})$. For a subset
$\mathcal{P}\subset\mathcal{M}$, the upper expectation of $\mathcal{P}$ is
defined as follows: for each random variable $X$,

$\mathbb{E}^{\mathcal{P}}[X]:=\sup_{P\in\mathcal{P}}E_{P}[X].$

It is easy to verify that $\mathbb{E}^{\mathcal{P}}$ is a sublinear
expectation, i.e., $\mathbb{E}^{\mathcal{P}}$ satisfies following properties:

  * (i)

Monotonicity: $\mathbb{E}^{\mathcal{P}}[X]\leq\mathbb{E}^{\mathcal{P}}[Y],$ if
$X\leq Y$,

  * (ii)

Constant preserving: $\mathbb{E}^{\mathcal{P}}[c]=c,\forall c\in\mathbb{R}$,

  * (iii)

Sub-additivity:
$\mathbb{E}^{\mathcal{P}}[X+Y]\leq\mathbb{E}^{\mathcal{P}}[X]+\mathbb{E}^{\mathcal{P}}[Y]$,

  * (iv)

Positive homogeneity: $\mathbb{E}^{\mathcal{P}}[\lambda
X]=\lambda\mathbb{E}^{\mathcal{P}}[X],\forall\lambda\geq 0$.

For each $A\in\mathcal{F}$, the upper and lower probabilities are defined
corresponding to the set of probability measures $\mathcal{P}$ respectively as
follows:

$V(A):=\sup_{P\in\mathcal{P}}P(A)$ $v(A):=\inf_{P\in\mathcal{P}}P(A).$

Obviously, $V(\cdot)$ and $v(\cdot)$ are conjugate to each other, that is

$V(A)+v\left(A^{c}\right)=1.$

The following notions of identical distribution and independence on the
sublinear expectation space are initiated by Peng [9] (see also Peng [8, 10]).

######  Definition 2.1.

Let $X$ and $Y$ be two random variables on sublinear expectation spaces
$(\Omega_{1},\mathcal{F}_{1},{\mathcal{P}_{1}})$ and
$(\Omega_{2},\mathcal{F}_{2},{\mathcal{P}_{2}})$ respectively. $X$ and $Y$ are
called identically distributed, if for each $\varphi\in
C_{b,{Lip}}(\mathbb{R})$,

$\mathbb{E}^{\mathcal{P}_{1}}[\varphi(X)]=\mathbb{E}^{\mathcal{P}_{2}}[\varphi(Y)],$

where $C_{b,Lip}(\mathbb{R})$ is the set of all bounded and Lipschitz
functions on $\mathbb{R}$.

It is worth pointing out that if $X$ and $Y$ are identical distributed by
Definition 2.1, we can not obtain that $V_{1}(X\in A)=V_{2}(Y\in A),\forall
A\in\mathcal{B}(\mathbb{R})$ in general, where $V_{i}$ is introduced by
$\mathcal{P}_{i},i=1,2.$

######  Example 2.2.

Let $(\Omega,\mathcal{F})=([0,1],\mathcal{B}[0,1])$,
$\mathcal{P}_{1}=\\{\delta_{x},x\in[0,1)\\}$,
$\mathcal{P}_{2}=\\{\delta_{x},x\in[0,1]\\}$, where $\delta_{x}$ is Dirac
measure at $x$. Consider the random variables $X$ and $Y$ defined by
$X(\omega)=Y(\omega)=\omega$. Then for each continuous function $\varphi\in
C_{b,Lip}(\mathbb{R})$,

$\mathbb{E}^{\mathcal{P}_{1}}[\varphi(X)]=\max_{x\in[0,1]}\varphi(x)=\mathbb{E}^{\mathcal{P}_{2}}[\varphi(Y)],$

but we have

$V_{1}(X=1)=0\neq 1=V_{2}(X=1).$

In particular, if we further assume that both $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ are weakly compact, we can deduce that $V_{1}(X\in
A)=V_{2}(Y\in A),\ \forall A\in\mathcal{B}(\mathbb{R})$, see Proposition 14 in
Hu and Peng [5] or Proposition 2.8 in Li [6].

######  Definition 2.3.

Let $\left\\{X_{n}\right\\}_{n=1}^{\infty}$ be a sequence of random variables
on $(\Omega,\mathcal{F},\mathcal{P})$. $X_{n}$ is said to be independent of
$\left(X_{1},\cdots,X_{n-1}\right)$ under $\mathcal{P}$, if for each
$\varphi\in C_{b,\operatorname{Lip}}\left(\mathbb{R}^{n}\right)$

$\mathbb{E}^{\mathcal{P}}\left[\varphi\left(X_{1},\cdots,X_{n}\right)\right]=\mathbb{E}^{\mathcal{P}}\left[\left.\mathbb{E}^{\mathcal{P}}\left[\varphi\left(x_{1},\cdots,x_{n-1},X_{n}\right)\right]\right|_{\left(x_{1},\cdots,x_{n-1}\right)=\left(X_{1},\cdots,X_{n-1}\right)}\right].$

The sequence of random variables $\left\\{X_{n}\right\\}_{n=1}^{\infty}$ is
said to be independent, if $X_{n+1}$ is independent of
$\left(X_{1},\cdots,X_{n}\right)$ for each $n\in\mathbb{N}$.

In the sublinear expectation space, there also exists $G$-normal distribution
which plays an important role as an analogue of the normal distribution in the
classical probability theory.

######  Definition 2.4.

A random variable $\xi$ on a sublinear expectation space
$(\Omega,\mathcal{F},\mathcal{P})$ is called $G$-normal distributed, denoted
by $\xi\sim\mathcal{N}(0;[\underline{\sigma}^{2},\overline{\sigma}^{2}])$, for
a given pair $0\leq\underline{\sigma}\leq\overline{\sigma}<\infty$, if for
each $\varphi\in C_{b,Lip}(\mathbb{R})$, the following function defined by

$u(t,x)=\mathbb{E}^{\mathcal{P}}[\varphi(x+\sqrt{t}\xi)],(t,x)\in[0,\infty)\times\mathbb{R},$

is the unique viscosity solution of the following parabolic partial
differential equation defined on $[0,\infty)\times\mathbb{R}$:

$\partial_{t}u-G(\partial^{2}_{xx}u)=0,u|_{t=0}=\varphi,$

where

$G(\alpha)=\frac{1}{2}(\overline{\sigma}^{2}\alpha^{+}-\underline{\sigma}^{2}\alpha^{-}),\alpha\in\mathbb{R}.$

In particular, if $\overline{\sigma}=\underline{\sigma}$, then $\xi$ defined
above is the classical normal distribution.

##  3\. Properties of pseudo-independent random variables

Recently, Guo and Li [4] (see also Li [6]) introduced the concept of pseudo-
independence, which characterizes the relations between classical conditional
expectation and sublinear expectation.

######  Definition 3.1.

Let $\left\\{X_{n}\right\\}_{n=1}^{\infty}$ be a sequence of random variables
on $(\Omega,\mathcal{F},\mathcal{P})$. Define

$\mathcal{F}_{n}:=\sigma\left(X_{1},\cdots,X_{n}\right),n\geq
1\quad\text{and}\quad\mathcal{F}_{0}:=\\{\emptyset,\Omega\\}.$

If for each $P\in\mathcal{P}$, we have

(3.1)
$E_{P}\left[\varphi\left(X_{n}\right)\mid\mathcal{F}_{n-1}\right]\leq\mathbb{E}^{\mathcal{P}}\left[\varphi\left(X_{n}\right)\right]\quad
P\text{-a.s.},\quad\forall\varphi\in C_{b,\operatorname{Lip}}(\mathbb{R}),$

then we call that $X_{n}$ is pseudo-independent of
$\left(X_{1},\cdots,X_{n-1}\right)$.

The pseudo-independent sequence $\\{X_{n}\\}_{n=1}^{\infty}$ can be defined in
the same manner as the independent sequence.

######  Remark 3.2.

It is worth pointing out that (3.1) is equivalent to

$-\mathbb{E}^{\mathcal{P}}\left[-\varphi\left(X_{n}\right)\right]\leq
E_{P}\left[\varphi\left(X_{n}\right)\mid\mathcal{F}_{n-1}\right]\leq\mathbb{E}^{\mathcal{P}}\left[\varphi\left(X_{n}\right)\right],\
\forall\varphi\in C_{b,\operatorname{Lip}}(\mathbb{R}).$

In particular, if $\mathcal{P}=\\{P\\}$ is a singleton set, as shown in the
classical probability theory, $X$ being independent of the $\sigma$-field
$\mathcal{F}$ is equivalent to
$E_{P}[\varphi(X)|\mathcal{F}]=E_{P}[\varphi(X)]$ for each test function
$\varphi\in C_{b,Lip}(\mathbb{R})$. However, pseudo-independence means that
the conditional expectation $E[\varphi(X)|\mathcal{F}]$ may not exactly equal
to the unconditional expectation $E_{P}[\varphi(X)]$, but lie in a non-random
interval depending on $\varphi$.

Here is an important property of these concepts, which has been proved in Guo
and Li [4].

######  Proposition 3.3.

An independent sequence of random variables
$\left\\{X_{n}\right\\}_{n=1}^{\infty}$ on the sublinear expectation space
$(\Omega,\mathcal{F},\mathcal{P})$ is pseudo-independent.

Pseudo-independence presents more extensive application scenarios than the
independence in sublinear expectation space proposed earlier. For instance,
independence under each probability $P$ of $\mathcal{P}$ apparently implies
pseudo-independence, but usually can not guarantee independence in Definition
2.3, see Example 3.6 in the end of this section. In general, we have the
following properties.

######  Proposition 3.4.

Let $\\{X_{n}\\}_{n=1}^{\infty}$ be a pseudo-independent sequence under
$\mathcal{P}$, then for each $n\in\mathbb{N}$, $\forall\varphi\in
C_{b,Lip}(\mathbb{R}^{n+1})$,

$\mathbb{E}^{\mathcal{P}}[\varphi(X_{1},\cdots,X_{n},X_{n+1})]\leq\mathbb{E}^{\mathcal{P}}[\mathbb{E}^{\mathcal{P}}[\varphi(x_{1},\cdots,x_{n},X_{n+1})]|_{(x_{1},\cdots,x_{n})=(X_{1},\cdots,X_{n})}].$

###### Proof.

It is clear that

$\displaystyle\mathbb{E}^{\mathcal{P}}[\varphi(X_{1},\cdots,X_{n},X_{n+1})]=$
$\displaystyle\sup_{P\in\mathcal{P}}E_{P}[\varphi(X_{1},\cdots,X_{n+1})]$
$\displaystyle=$
$\displaystyle\sup_{P\in\mathcal{P}}E_{P}[E_{P}[\varphi(x_{1},\cdots,x_{n},X_{n+1})|\mathcal{F}_{n}]|_{(x_{1},\cdots,x_{n})=(X_{1},\cdots,X_{n})}]$
$\displaystyle\leq$
$\displaystyle\sup_{P\in\mathcal{P}}E_{P}[\mathbb{E}^{\mathcal{P}}[\varphi(x_{1},\cdots,x_{n},X_{n+1})]_{(x_{1},\cdots,x_{n})=(X_{1},\cdots,X_{n})}]$
$\displaystyle=$
$\displaystyle\mathbb{E}^{\mathcal{P}}[\mathbb{E}^{\mathcal{P}}[\varphi(x_{1},\cdots,x_{n},X_{n+1})]|_{(x_{1},\cdots,x_{n})=(X_{1},\cdots,X_{n})}].$

∎

Proposition 3.4 inspires us to enlarge $\mathcal{P}$ to $\bar{\mathcal{P}}$
such that $\\{X_{n}\\}_{n=1}^{\infty}$ is independent under
$\bar{\mathcal{P}}$.

######  Proposition 3.5.

Let $\\{X_{n}\\}_{n=1}^{\infty}$ be a pseudo-independent sequence under
$\mathbb{E}^{\mathcal{P}}$. We consider $\bar{\mathcal{P}}$ defined by

$\bar{\mathcal{P}}=\\{P\in\mathcal{M}:\forall\varphi\in
C_{b,Lip}(\mathbb{R}),E_{P}[\varphi(X_{n})|\mathcal{F}_{n-1}]\leq\mathbb{E}^{\mathcal{P}}[\varphi(X_{n})],\forall
n\in\mathbb{N}\\}$

then we have $\bar{\mathcal{P}}\supset\mathcal{P}$ and
$\\{X_{n}\\}_{n=1}^{\infty}$ is independent under
$\mathbb{E}^{\bar{\mathcal{P}}}$ with
$\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(X_{n})]=\mathbb{E}^{\mathcal{P}}[\varphi(X_{n})],\forall
n\in\mathbb{N},\forall\varphi\in C_{b,Lip}(\mathbb{R})$.

Moreover, $\forall n\in\mathbb{N}$,

$\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(X_{1},\cdots,X_{n+1})]=\mathbb{E}^{\mathcal{P}}[\mathbb{E}^{\mathcal{P}}[\varphi(x_{1},\cdots,x_{n},X_{n+1})]|_{(x_{1},\cdots,x_{n})=(X_{1},\cdots,X_{n})}].$

###### Proof.

For simplicity, we only consider the case of $n=2$. We consider

$\bar{\mathcal{P}}:=\\{P:\forall\varphi\in
C_{b,Lip}(\mathbb{R}),E_{P}[\varphi(X_{1})]\leq\mathbb{E}^{\mathcal{P}}[\varphi(X_{1})]\
\text{and}\
E_{P}[\varphi(X_{2})|X_{1}]\leq\mathbb{E}^{\mathcal{P}}[\varphi(X_{2})]\\}.$

In this case, for each fixed $\varphi\in C_{b,Lip}(\mathbb{R}^{2})$ and
$\varepsilon>0$, there exits $P^{*}\in\bar{\mathcal{P}}$ (depending on
$\varphi$ and $\varepsilon$) such that

(i)
$E_{P^{*}}[\psi(X_{1})]\geq\mathbb{E}^{\mathcal{P}}[\psi(X_{1})]-\frac{\varepsilon}{2}$,
where $\psi(x)=\mathbb{E}^{\mathcal{P}}[\varphi(x,X_{2})]$.

(ii)
$E_{P^{*}}[\varphi(x,X_{2})|X_{1}]\geq\mathbb{E}^{\mathcal{P}}[\varphi(x,X_{2})]-\frac{\varepsilon}{2}$,
for all $x$ belongs to the range of $X_{1}$.

Then we have

$\displaystyle\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(X_{1},X_{2})]\geq$
$\displaystyle E_{P^{*}}[\varphi(X_{1},X_{2})]$ $\displaystyle=$
$\displaystyle E_{P^{*}}[E_{P^{*}}[\varphi(x,X_{2})|X_{1}]_{x=X_{1}}]$
$\displaystyle\geq$ $\displaystyle
E_{P^{*}}[\mathbb{E}^{\mathcal{P}}[\varphi(x,X_{2})]|_{x=X_{1}}]-\frac{\varepsilon}{2}$
$\displaystyle\geq$
$\displaystyle\mathbb{E}^{\mathcal{P}}[\mathbb{E}^{\mathcal{P}}[\varphi(x,X_{2})]|_{x=X_{1}}]-\varepsilon$
$\displaystyle=$
$\displaystyle\mathbb{E}^{\bar{\mathcal{P}}}[\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(x,X_{2})]|_{x=X_{1}}]-\varepsilon.$

Combining with Proposition 3.4 and noticing that $\varepsilon$ can be
arbitrarily small, we see that $X_{2}$ is independent of $X_{1}$ under
$\bar{\mathcal{P}}$. ∎

We conclude this section by an example.

######  Example 3.6.

We consider a probability set $\mathcal{P}=\\{P_{1},P_{2}\\}$ and two binary
random variables $X$ and $Y$ satisfying

$\displaystyle P_{1}(X=0,Y=0)=\frac{1}{16},\ P_{1}(X=1,Y=1)=\frac{9}{16}$
$\displaystyle P_{1}(X=1,Y=0)=P_{1}(X=0,Y=1)=\frac{3}{16}$ $\displaystyle
P_{2}(X=0,Y=0)=P_{2}(X=1,Y=1)=P_{2}(X=0,Y=1)=P_{2}(X=1,Y=0)=\frac{1}{4}$

We can see that $X$ and $Y$ are independent of each other under $P_{1}$ and
$P_{2}$ respectively, it is clear that $Y$ is pseudo-independent of $X$ under
$\mathcal{P}$. Let $\varphi^{*}$ be a bonded and Lipschitz function with

(3.2)
$\varphi^{*}(0,0)=\varphi^{*}(1,1)=1,\quad\varphi^{*}(1,0)=\varphi^{*}(0,1)=0.$

By the simple calculation, we have

$\mathbb{E}^{\mathcal{P}}\left[\varphi^{*}(X,Y)\right]=\frac{5}{8}<\frac{11}{16}=\mathbb{E}^{\mathcal{P}}\left[\mathbb{E}^{\mathcal{P}}\left[\varphi^{*}(x,Y)\right]|_{x=X}\right],$

which indicates that $Y$ is not independent of $X$.

Now we enlarge $\mathcal{P}$ to $\bar{\mathcal{P}}$ by

$\bar{\mathcal{P}}:=\\{P:\forall\varphi\in
C_{b,Lip}(\mathbb{R}),E_{P}[\varphi(X)]\leq\mathbb{E}^{\mathcal{P}}[\varphi(X)]\
\text{and}\ E_{P}[\varphi(Y)|X]\leq\mathbb{E}^{\mathcal{P}}[\varphi(Y)]\\}.$

In particular, for $\varphi^{*}$ defined in (3.2), we can choose $P^{*}$ by

$\displaystyle P^{*}(X=1,Y=1)=\frac{9}{16}\ \ \ $ $\displaystyle
P^{*}(X=1,Y=0)=\frac{3}{16}$ $\displaystyle P^{*}(X=0,Y=1)=\frac{1}{8}\ \ \ $
$\displaystyle P^{*}(X=0,Y=0)=\frac{1}{8}$

It is easily to verify that $P^{*}\notin\mathcal{P}$ but
$P^{*}\in\bar{\mathcal{P}}$, Furthermore, we have

$\displaystyle\mathbb{E}^{\bar{\mathcal{P}}}[\mathbb{E}^{\bar{\mathcal{P}}}\left[\varphi^{*}(x,Y)]|_{x=X}\right]$
$\displaystyle=E_{P^{*}}\left[E_{P^{*}}\left[\varphi^{*}(x,Y)\right]|_{x=X}\right]$
$\displaystyle=\frac{11}{16}=E_{P^{*}}[\varphi^{*}(X,Y)]=\mathbb{E}^{\bar{\mathcal{P}}}[\varphi^{*}(X,Y)].$

##  4\. Law of large numbers for pseudo-independent random variables

We firstly consider the law of large numbers for pseudo-independent sequence.

Let $\\{X_{n}\\}_{n=1}^{\infty}$ be a sequence of random variables on the
sublinear expectation space $(\Omega,\mathcal{F},\mathcal{P}),$ and denote
$S_{n}=X_{1}+\cdots+X_{n}$, $S_{0}=0$,
$\mathcal{F}_{n}=\sigma(X_{1},X_{2},...,X_{n})$ for $n\geq 1$ and
$\mathcal{F}_{0}=\\{\emptyset,\Omega\\}$. In this section, we require the
sequence $\\{X_{n}\\}_{n=1}^{\infty}$ consistent with the following
hypothesis:

  * (H1)

There exist a random variable $X$ with $\lim_{n\to\infty}nV(|X|\geq n)=0$ and
a constant $K$ such that $V\left(|X_{n}|\geq x\right)\leq KV\left(|X|\geq
x\right)$ for each $x\geq 0$ and $n\geq 1.$

We have the following law of large numbers for pseudo-independent sequence.

######  Theorem 4.1.

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be a pseudo-independent sequence under
$\mathcal{P}$ satisfying (H1), we define

$\overline{\mu}:=\limsup_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[X_{i}\mathbf{I}_{\\{|X_{i}|<n\\}}]}{n}\
\ \
\underline{\mu}:=\limsup_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}-\mathbb{E}^{\mathcal{P}}[-X_{i}\mathbf{I}_{\\{|X_{i}|<n\\}}]}{n}$

then we have $\forall\varphi\in C_{b,Lip}(\mathbb{R})$,

(4.1)
$\min_{\underline{\mu}\leq\mu\leq\overline{\mu}}\varphi(\mu)\leq\liminf_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{n})]\leq\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{n})]\leq\max_{\underline{\mu}\leq\mu\leq\overline{\mu}}\varphi(\mu).$

To complete the proof, we need some lemmas. The first lemma is Lemma 3.1 in
Guo and Li [4].

######  Lemma 4.2.

Let $\\{X_{i}\\}_{i=1}^{\infty}$ satisfy (H1), then

(4.2)
$\lim_{n\rightarrow\infty}\frac{1}{n^{2}}\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[|X_{i}|^{2}\mathbf{I}_{\\{|X_{i}|\leq
n\\}}]=0.$

The next lemma generalizes the corresponding result for i.i.d. sequence in
Chen et al. [1] to the pseudo-independent sequence. Here we give a very simple
proof.

######  Lemma 4.3.

Under the same condition of Theorem 4.1, we have, for any $\varepsilon>0$,

$\lim_{n\to\infty}v(\underline{\mu}-\varepsilon\leq\frac{S_{n}}{n}\leq\overline{\mu}+\varepsilon)=1.$

###### Proof.

Let $Y_{n,i}=X_{i}\mathbf{I}_{\\{|X_{i}|<n\\}}$ and
$T_{n}=\sum_{i=1}^{n}Y_{n,i}$. For each $P\in\mathcal{P}$, obviously,
$E_{P}[Y_{n,i}^{2}]\leq\mathbb{E}^{\mathcal{P}}[X_{i}^{2}]$, and
$-\mathbb{E}^{\mathcal{P}}[-Y_{n,i}]\leq
E_{P}[Y_{n,i}|\mathcal{F}_{i-1}]\leq\mathbb{E}^{\mathcal{P}}[Y_{n,i}]$.

Firstly, by Chebyshev’s inequality and Jensen’s inequality, we have

$\displaystyle
P(\frac{\sum_{i=1}^{n}(Y_{n,i}-E_{P}[Y_{n,i}|\mathcal{F}_{i-1}])}{n}>\varepsilon)$
$\displaystyle\leq\frac{\sum_{i=1}^{n}(E_{P}[Y_{n,i}^{2}]+E_{P}[(E_{P}[Y_{n,i}|\mathcal{F}_{i-1}])^{2}])}{n^{2}\varepsilon^{2}}$
$\displaystyle\leq\frac{2\sum_{i=1}^{n}E_{P}[Y_{n,i}^{2}]}{n^{2}\varepsilon^{2}}\leq\frac{2\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[Y_{n,i}^{2}]}{n^{2}\varepsilon^{2}}.$

Noting that,

$P(S_{n}\neq T_{n})\leq\sum_{i=1}^{n}P(|X_{i}|\geq
n)\leq\sum_{i=1}^{n}V(|X_{i}|\geq n)\leq nKV(|X|\geq n),$

thus

$\displaystyle P(\frac{S_{n}}{n}>\overline{\mu}+\varepsilon)$
$\displaystyle\leq P(S_{n}\neq
T_{n})+P(\frac{\sum_{i=1}^{n}(Y_{n,i}-E[Y_{n,i}|\mathcal{F}_{i-1}])}{n}>\frac{\varepsilon}{2})$
$\displaystyle\ \ \ \
+P(\frac{\sum_{i=1}^{n}E[Y_{n,i}|\mathcal{F}_{i-1}]}{n}>\overline{\mu}+\frac{\varepsilon}{2})$
$\displaystyle\leq nKV(|X|\geq
n)+\frac{8\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[Y_{n,i}^{2}]}{n^{2}\varepsilon^{2}}+P(\frac{\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[Y_{n,i}]}{n}>\overline{\mu}+\frac{\varepsilon}{2}).$

Combining with Lemma 4.2, we can imply that

$V(\frac{S_{n}}{n}>\overline{\mu}+\varepsilon)\rightarrow 0,$

Similarly, we can prove that
$V(\frac{S_{n}}{n}<\underline{\mu}-\varepsilon)\rightarrow 0$, the proof is
completed. ∎

Now we give the proof of Theorem 4.1.

###### Proof of Theorem 4.1.

If we take $\varphi^{\prime}=-\varphi$ in (4.1), we only need to prove that

$\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{n})]\leq\max_{\underline{\mu}\leq\mu\leq\overline{\mu}}\varphi(\mu).$

For each $\varepsilon>0$, we have

$\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{n})]\leq\max_{\underline{\mu}-\varepsilon\leq\mu\leq\overline{\mu}+\varepsilon}\varphi(\mu)+\max_{\mu\in\mathbb{R}}\varphi(\mu)(V(\frac{S_{n}}{n}>\overline{\mu}+\varepsilon)+V(\frac{S_{n}}{n}<\underline{\mu}-\varepsilon)).$

Since $\varphi$ is bounded, letting $n\rightarrow\infty$, we can complete the
proof. ∎

We reformulate Peng’s LLN (see Theorem 2.4.1 in Peng [10]) for the probability
set $\mathcal{P}$ and give a purely probabilistic proof. Such probabilistic
proof was firstly considered in Li [6] (see Theorem 2.2) under the condition
of $\mathbb{E}^{\mathcal{P}}[|X_{1}|^{1+\delta}]<\infty$ for some $\delta>0$.
The current proof is totally different from the original one in Peng [10]
based on the PDE estimation.

######  Theorem 4.4.

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be an i.i.d. sequence under
$\mathbb{E}^{\mathcal{P}}$ with
$\lim_{\lambda\rightarrow+\infty}\mathbb{E}^{\mathcal{P}}[(|X_{1}|-\lambda)^{+}]=0$,
then

(4.3)
$\lim_{n\rightarrow\infty}\mathbb{E}[\varphi(\frac{X_{1}+\cdots+X_{n}}{n})]=\max_{-\mathbb{E}[-X_{1}]\leq\mu\leq\mathbb{E}[X_{1}]}\varphi(\mu),$

where $\varphi$ is any continuous function satisfying linear growth condition.

###### Proof.

We only need to prove that (4.3) holds for $\varphi\in C_{b,Lip}(\mathbb{R})$.
By the same argument in Peng [10], it can be easily generalized to $\varphi$
with linear growth condition.

It is clear that $nV(|X_{1}|\geq n)\rightarrow 0$ and
$\mathbb{E}^{\mathcal{P}}[\pm
X_{1}\mathbf{I}_{\\{|X_{1}|<n\\}}]\rightarrow\mathbb{E}^{\mathcal{P}}[\pm
X_{1}]$ since
$\lim_{\lambda\rightarrow+\infty}\mathbb{E}^{\mathcal{P}}[(|X_{1}|-\lambda)^{+}]=0$.
Then by Theorem 4.1, we have

$\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{n})]\leq\max_{-\mathbb{E}^{\mathcal{P}}[-X_{1}]\leq\mu\leq\mathbb{E}^{\mathcal{P}}[X_{1}]}\varphi(\mu).$

On the other hand, for fixed $\varphi\in C_{b,Lip}(\mathbb{R})$ with Lipschitz
constant $L_{\varphi}$ and bound $C_{\varphi}$, and for each
${\varepsilon}>0$, there exists
$\mu\in(-\mathbb{E}^{\mathcal{P}}[-X_{1}],\mathbb{E}^{\mathcal{P}}[X_{1}])$
such that
$\varphi(\mu)\geq\max_{-\mathbb{E}^{\mathcal{P}}[-X_{1}]\leq\mu\leq\mathbb{E}^{\mathcal{P}}[X_{1}]}\varphi(\mu)-{\varepsilon}$,
we can find $E_{P}$ such that $\\{X_{i}\\}$ is an i.i.d. sequence under
$E_{P}$ with $E_{P}[X_{1}]=\mu$, where $P$ can be chosen from
$\bar{\mathcal{P}}$ defined in Proposition 3.5. In this case, by the classical
weak law of large numbers,

$P(|\frac{S_{n}}{n}-\mu|>\varepsilon)\rightarrow 0.$

Then

$|E_{P}[\varphi(\frac{S_{n}}{n})]-\varphi(\mu)|\leq
L_{\varphi}\varepsilon+2C_{\varphi}P(|\frac{S_{n}}{n}-\mu_{n}|>\varepsilon),$

We can obtain that

$\lim_{n\rightarrow\infty}|E_{P}[\varphi(\frac{S_{n}}{n})]-\varphi(\mu)|=0,$

which implies that, for each ${\varepsilon}>0$,

(4.4)
$\liminf_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{n})]\geq\max_{-\mathbb{E}^{\mathcal{P}}[-X_{1}]\leq\mu\leq\mathbb{E}^{\mathcal{P}}[X_{1}]}\varphi(\mu)-\varepsilon.$

Let $\varepsilon\rightarrow 0$, the LLN holds. ∎

##  5\. Central limit theorem for pseudo-independent random variables

In this section, we consider the CLT for pseudo-independent sequence. We
firstly reformulate the Peng’s CLT in Peng [10] on sublinear expectation space
$(\Omega,\mathcal{F},\mathcal{P})$. The further result, for example, the
convergence rate of CLT can be found in Fang et al. [3] and Song [11].

######  Theorem 5.1.

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be an i.i.d. sequence of random variables on
$(\Omega,\mathcal{F},\mathcal{P})$, we assume further

$\lim_{\lambda\rightarrow+\infty}\mathbb{E}^{\mathcal{P}}[(|X_{1}|^{2}-\lambda)^{+}]=0,$

then

$\lim_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}})]=\mathbb{E}^{\mathcal{P}}[\varphi(\xi)],$

for all continuous function $\varphi$ with linear growth condition, where
$\xi$ is a $G$-normally distributed random variable with
$\xi\sim\mathcal{N}(0,[-\mathbb{E}^{\mathcal{P}}[-X_{1}^{2}],\mathbb{E}^{\mathcal{P}}[X_{1}^{2}]])$.

Recently, Zhang [12] has weaken the condition
$\lim_{\lambda\rightarrow+\infty}\mathbb{E}[(|X_{1}|^{2}-\lambda)^{+}]=0$ to
$nV(|X_{1}|^{2}\geq n)\rightarrow 0$, which inspires us to assume the sequence
$\\{X_{n}\\}_{n=1}^{\infty}$ consistent with the following hypothesis:

  * (H2)

There exist a random variable $X$ with $\lim_{n\to\infty}nV(X^{2}\geq n)=0$
and a constant $K$ such that $V\left(|X_{n}|\geq x\right)\leq KV\left(|X|\geq
x\right)$ for each $x\geq 0$ and $n\geq 1.$

Let $S_{n}=X_{1}+\cdots+X_{n}$, we have the following central limit theorem
for pseudo-independent sequence.

######  Theorem 5.2.

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be a pseudo-independent sequence under
$\mathcal{P}$ satisfying (H2) and
$\mathbb{E}^{\mathcal{P}}[X_{i}]=\mathbb{E}^{\mathcal{P}}[-X_{i}]=0,\forall
i\in\mathbb{N}$, we define

$\overline{\sigma}^{2}:=\limsup_{i\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[X_{i}^{2}],\
\ \
\underline{\sigma}^{2}:=\liminf_{i\rightarrow\infty}-\mathbb{E}^{\mathcal{P}}[-X_{i}^{2}].$

We further assume that $\overline{\sigma}<+\infty$, then we have,
$\forall\varphi\in C_{b,Lip}(\mathbb{R})$,

$-\mathbb{E}^{\mathcal{P}}[-\varphi(\xi)]\leq\liminf_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\mathbb{E}^{\mathcal{P}}[\varphi(\xi)],$

where $\xi$ is a $G$-normally distributed random variable with
$\xi\sim\mathcal{N}(0,[\underline{\sigma}^{2},\overline{\sigma}^{2}])$.

To prove Theorem 5.2, we need the following lemmas.

######  Lemma 5.3.

Let $\\{X_{ij}\\}_{1\leq j\leq i}^{\infty}$ be the double array, where for
each row, $\\{X_{ni}\\}_{i=1}^{n}$ is independent sequence under
$\mathbb{E}^{\mathcal{P}}$. We assume that

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\mathbb{E}^{\mathcal{P}}[|X_{ni}|]+\mathbb{E}^{\mathcal{P}}[-X_{ni}])\rightarrow
0,$

and

$\frac{1}{n^{3/2}}\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[|X_{ni}|^{3}]\rightarrow
0.$

Define $\overline{\sigma}^{*}=\sup_{i,j}\mathbb{E}^{\mathcal{P}}[X_{ij}^{2}]$
and $\underline{\sigma}^{*}=\inf_{i,j}-\mathbb{E}^{\mathcal{P}}[-X_{ij}^{2}]$,
we further assume that $\overline{\sigma}<+\infty$. Then we have

$-\mathbb{E}^{\mathcal{P}}[-\varphi(\xi^{*})]\leq\liminf_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\mathbb{E}^{\mathcal{P}}[\varphi(\xi^{*})],$

where $\xi$ is a $G$-normally distributed random variable with
$\xi^{*}\sim\mathcal{N}(0,[{\underline{\sigma}^{*}}^{2},{\overline{\sigma}^{*}}^{2}])$.

The proof is very similar to the corresponding one in Li [7], so we omit the
proof here.

######  Lemma 5.4.

Let $\\{X_{i}\\}_{i=1}^{\infty}$ be an independent sequence under
$\mathbb{E}^{\mathcal{P}}$ satisfying (H2) and
$\mathbb{E}^{\mathcal{P}}[X_{i}]=\mathbb{E}^{\mathcal{P}}[-X_{i}]=0,\forall
i\in\mathbb{N}$. Define
$\overline{\sigma}^{*}=\sup_{i}\mathbb{E}^{\mathcal{P}}[X_{i}^{2}]$ and
$\underline{\sigma}^{*}=\inf_{i}-\mathbb{E}^{\mathcal{P}}[-X_{i}^{2}]$, we
further assume that $\overline{\sigma}<+\infty$. Then we have

$-\mathbb{E}^{\mathcal{P}}[-\varphi(\xi^{*})]\leq\liminf_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\mathbb{E}^{\mathcal{P}}[\varphi(\xi^{*})],$

where $\xi$ is a $G$-normally distributed random variable with
$\xi^{*}\sim\mathcal{N}(0,[{\underline{\sigma}^{*}}^{2},{\overline{\sigma}^{*}}^{2}])$.

###### Proof.

Let $X_{ni}=(-\sqrt{n})\vee X_{i}\wedge\sqrt{n}$, by the same argument in
Zhang [12], $\\{X_{ni}\\}$ satisfies the conditions in Lemma 5.3, we have

$\limsup_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{\sum_{i=1}^{n}X_{ni}}{\sqrt{n}})]\leq\mathbb{E}^{\mathcal{P}}[\varphi(\xi^{*})].$

It is clear that

$\mathbb{E}^{\mathcal{P}}[|\varphi(\frac{\sum_{i=1}^{n}X_{ni}}{\sqrt{n}})-\varphi(\frac{S_{n}}{\sqrt{n}})|]\leq\sup_{x}|\varphi(x)|\sum_{i=1}^{n}V(|X_{i}|\geq\sqrt{n})\leq
KnV(|X|\geq\sqrt{n})\rightarrow 0,$

we can complete the proof. ∎

Now we give the proof of Theorem 5.2.

###### Proof of Theorem 5.2.

By Proposition 3.5, we can enlarge $\mathcal{P}$ to $\bar{\mathcal{P}}$ such
that $\\{X_{i}\\}_{i=1}^{\infty}$ is independent under
$\mathbb{E}^{\bar{\mathcal{P}}}$ with $\mathbb{E}^{\bar{\mathcal{P}}}[\pm
X_{i}^{p}]=\mathbb{E}^{\mathcal{P}}[\pm X_{i}^{p}]$, $p=1,2$. We further
assume that $\underline{\sigma}>0$. In fact, this condition can be removed by
the perturbation method in the proof of Theorem 2.4.7 in Peng [10].

For each $0<\varepsilon<\underline{\sigma}$, we can choose $N$ such that
$\mathbb{E}^{\bar{\mathcal{P}}}[X_{i}^{2}]\leq\overline{\sigma}^{2}+\varepsilon$
and
$-\mathbb{E}^{\bar{\mathcal{P}}}[-X_{i}^{2}]\geq\underline{\sigma}^{2}-\varepsilon$,
when $i\geq N$. Set $S_{n}^{N}=\sum_{i=N}^{n}X_{i}$, for $n\geq N$, by Lemma
5.4, we have

$\limsup_{n\rightarrow\infty}\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(\frac{S_{n}^{N}}{\sqrt{n-N}})]\leq\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(\xi_{\varepsilon})],$

where
$\xi_{\varepsilon}\sim\mathcal{N}(0,[\underline{\sigma}^{2}-\varepsilon,\overline{\sigma}^{2}+\varepsilon])$.

It is easily seen that

$\displaystyle|\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(\frac{S_{n}}{\sqrt{n}})-\varphi(\frac{S_{n}^{N}}{n-N})]|$
$\displaystyle\leq\mathbb{E}^{\bar{\mathcal{P}}}[|\varphi(\frac{S_{n}}{\sqrt{n}})-\varphi(\frac{S_{n}^{N}}{\sqrt{n}})|]+\mathbb{E}^{\bar{\mathcal{P}}}[|\varphi(\frac{S_{n}^{N}}{\sqrt{n}})-\varphi(\frac{S_{n}^{N}}{\sqrt{n-N}})|]$
$\displaystyle\leq
C\frac{\sum_{i=1}^{N}\mathbb{E}^{\bar{\mathcal{P}}}[|X_{i}|]}{\sqrt{n}}+C\frac{N\sum_{i=N}^{n}\mathbb{E}^{\bar{\mathcal{P}}}[|X_{i}|]}{\sqrt{n(n-N)(\sqrt{n}+\sqrt{n+N})}}\rightarrow
0,$

Thus we obtain

$\limsup_{n\rightarrow\infty}\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\mathbb{E}^{\mathcal{P}}[\varphi(\xi_{\varepsilon})].$

Note that
$\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]\leq\mathbb{E}^{\bar{\mathcal{P}}}[\varphi(\frac{S_{n}}{\sqrt{n}})]$,
and
$\mathbb{E}^{\mathcal{P}}[\varphi(\xi_{\varepsilon})]\rightarrow\mathbb{E}^{\mathcal{P}}[\varphi(\xi)]$
as $\varepsilon\rightarrow 0$, the proof is complete. ∎

##  6\. Example

In this section, we will present relevant counterexamples to illustrate that
the LLN and CLT for i.i.d sequence may fail only with
$\mathbb{E}^{\mathcal{P}}[|X_{1}|]<\infty$ and
$\mathbb{E}^{\mathcal{P}}[X_{1}^{2}]<\infty$ respectively.

######  Example 6.1.

Let $\Omega=\mathbb{Z}$, $\mathcal{F}=\mathcal{B}(\mathbb{Z})$,
$\mathcal{P}=\\{P_{k},k\geq 1\\}$, where $P_{k}(\\{0\\})=1-\frac{1}{k^{2}}$,
$P_{k}(\\{k\\})=P_{k}(\\{-k\\})=\frac{1}{2k^{2}}$. Consider a function $X$ on
$\mathbb{Z}$ defined by $X(n)=n,n\in\mathbb{Z}$. It is obvious that
$\mathbb{E}^{\mathcal{P}}[X]=\mathbb{E}^{\mathcal{P}}[-X]=0$ and
$\mathbb{E}^{\mathcal{P}}[X^{2}]=-\mathbb{E}^{\mathcal{P}}[-X^{2}]=1$.
According to Proposition 3.5 or Peng [10], we are able to construct an i.i.d.
sequence $\\{X_{n}\\}_{n=1}^{\infty}$ under $\mathbb{E}^{\mathcal{P}}$ such
that $X_{n}$ has the same distribution with $X$.

We have the following properties for such example.

######  Proposition 6.2.

In Example 6.1, let $Y_{i}=X_{i}^{2}$ and $\varphi(x)=1-x$, then

$\lim_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{Y_{1}+\cdots+Y_{n}}{n})]=1.$

###### Proof.

It is clear that,

$\displaystyle\mathbb{E}^{\mathcal{P}}[\varphi(\frac{x+Y_{i}}{n})]$
$\displaystyle=\sup_{k\in\mathbb{N}}\\{(1-\frac{1}{k^{2}})\varphi(\frac{x}{n})+\frac{1}{k^{2}}(\varphi(\frac{x+k^{2}}{n})\\}$
$\displaystyle=\sup_{k\in\mathbb{N}}\\{\varphi(\frac{x}{n})+\frac{1}{k^{2}}(\varphi(\frac{x+k^{2}}{n})-\varphi(\frac{x}{n}))\\}=\varphi(\frac{x}{n}).$

Then

$\displaystyle\mathbb{E}^{\mathcal{P}}[\varphi(\frac{Y_{1}+\cdots+Y_{n}}{n})]$
$\displaystyle=\mathbb{E}^{\mathcal{P}}[\mathbb{E}^{\mathcal{P}}[\varphi(\frac{x+Y_{n}}{n})]|_{x=Y_{1}+\cdots+Y_{n-1}}]$
$\displaystyle=\mathbb{E}^{\mathcal{P}}[\varphi(\frac{Y_{1}+\cdots+Y_{n-1}}{n})]=\cdots=\mathbb{E}^{\mathcal{P}}[\varphi(\frac{Y_{1}}{n})]=\varphi(0)=1.$

∎

Proposition 6.2 indicates that the Peng’s LLN (see Theorem 4.4) does not hold
only with the finite moment condition. Indeed,
$\lim_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]=1\neq
0=\varphi(1)$.

######  Proposition 6.3.

In Example 6.1, let $\varphi(x)=1-|x|$, then for all $n\in\mathbb{N}$,

(6.1)
$\displaystyle\mathbb{E}^{\mathcal{P}}[\varphi(\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}})]=1,$

thus the central limit theorem does not hold.

###### Proof.

We first observe that

$\displaystyle\mathbb{E}^{\mathcal{P}}[\varphi(\frac{x+X}{\sqrt{n}})]$
$\displaystyle=\sup_{k\in\mathbb{N}}\\{(1-\frac{1}{k^{2}})\varphi(\frac{x}{\sqrt{n}})+\frac{1}{2k^{2}}(\varphi(\frac{x-k}{\sqrt{n}})+\varphi(\frac{x+k}{\sqrt{n}}))\\}$
$\displaystyle=\varphi(\frac{x}{\sqrt{n}})+\frac{1}{\sqrt{n}}\sup_{k\in\mathbb{N}}\\{\frac{2|x|-|x+k|-|x-k|}{2k^{2}}\\}=\varphi(\frac{x}{\sqrt{n}}).$

Then

$\displaystyle\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]$
$\displaystyle=\mathbb{E}^{\mathcal{P}}[\mathbb{E}^{\mathcal{P}}[\varphi(\frac{x+X_{n}}{\sqrt{n}})]|_{x=X_{1}+\cdots+X_{n-1}}]=\mathbb{E}^{\mathcal{P}}[\varphi(\frac{X_{1}+\cdots+X_{n-1}}{\sqrt{n}})]$
$\displaystyle=\cdots=\mathbb{E}^{\mathcal{P}}[\varphi(\frac{X_{1}}{\sqrt{n}})]=\varphi(0)=1.$

In this case,
$\lim_{n\rightarrow\infty}\mathbb{E}^{\mathcal{P}}[\varphi(\frac{S_{n}}{\sqrt{n}})]=1>\mathcal{N}(\varphi)$,
where $\mathcal{N}(\varphi)$ is the expectation of $\varphi(\xi)$ and $\xi$ is
standard normal distributed. Peng’s CLT (Theorem 5.1) is invalid. ∎

In the end of the paper, we raise the following questions as the open
problems:

Questions Do the limit distributions of $\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$
and $\frac{X_{1}^{2}+\cdots+X_{n}^{2}}{n}$ in Example 6.1 (or in general case)
exist? If they exist, what are the corresponding limit distributions?

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