The laws of large numbers for Pareto-type random variables under sub-linear expectation

Abstract

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.

Keywords

Sub-linear expectation / Pareto type distribution / laws of large numbers / independent and identical distribution

1 Introduction

Since the beginning of the 20th century, significant progress has been made in the traditional theory of probability spaces, which has driven the development of several fields such as mathematical statistics and econometrics. Both probabilities and expectations are additive in traditional probability spaces, but most problems in the real world are not additive. Therefore, there is a need to consider a new framework that can be applied to analyze the hidden probability and statistical distribution uncertainties behind real-world problems such as risk management and quantum mechanics. Therefore, Peng [10-12] introduced the concept of sub-linear expectation space. The probability and expectation under the probability space are transformed into capacity and sub-linear expectation, and the phenomena that do not possess additivity in various domains are explained by capacity and sub-linear expectation, which remedies the shortcomings of the traditional probabilistic framework applied in uncertainty problems. Since then, literature on the theory and applications related to sub-linear expectation spaces has emerged, enriching and generalizing the corresponding results in traditional probability spaces. For example, Zhang [16-18] established a series of important inequalities such as the exponential inequality, Rosenthal's inequality, and the strong law of large numbers (SLLN).

In classical probability spaces, the research on weak law of large numbers (WLLN) and SLLN has been abundant and has gained important applications in many fields. The limit theory under the traditional probability space has become more and more mature, but under the sub-linear expectation space, the research difficulty increases because the capacity and sub-linear expectation are not additive, and at the same time many moment inequalities and exponential inequalities are difficult to establish, so the research of limit theory is very challenging. In recent years, more and more scholars focus on the study of WLLN and SLLN under sub-linear expectation space, and the theoretical results are becoming more and more abundant. Representative studies such as Chen [3] obtained the WLLN in the sub-linear expectation space with respect to a sequence of independent random variables; Peng [13], Zhang [17] obtained the Kolmogorov SLLN under different conditions; Chen et al. [4,5] obtained three different SLLNs under the sub-linear expectation space. Based on Chen [4], Hu [6,7] extended some corresponding results under general moment conditions. Wu and Jiang [14] conducted a systematic study of SLLN under sub-linear expectations and extended the results of Chen [4], Hu [6], Marinacci [9], and Zhang [17] to the general case. Ma and Wu [8] obtained SLLNs for weighted sums of sequences of extended negatively dependent (END) random variables in sub-linear expectation space using the method of Wu and Jiang [14].

In the classical probability space, Yang et al. [15] studied and established the laws of large numbers for Pareto-type random variables and obtained some WLLN and SLLN for the weighted sum of negatively superadditive-dependent (NSD) random variables. Similarly, in the sub-linear expectation space, we can also establish some laws of large numbers for random variables satisfying the Pareto distribution. Thus, the relevant conclusions in the classical probability space can be generalized to the sublinear expectation space. Specifically, for all x>0, we study the problem of convergence in capacity that satisfies the weighted sum of a random variables sequence obeying the Pareto distribution

(1.1)\mathbb{V}\left(X_n>x\right)=\frac{1}{x+c_n},

where \left\{c_n\right\} is a non-decreasing constant sequence, and c_n⩾1, which satisfies:

(1.2)C_n:=\sum _{j=1}^n\frac{1}{c_j}\to \mathrm{\infty }.

The symbol “\to” represents the limit whenn\to \mathrm{\infty }.

Moreover, Yang et al. [15] studied SLLN mainly considering the tails of the Pareto probability distribution. Similarly, we also mainly utilize the form of the tail of the Pareto tolerance when we study the convergence in capacity almost everywhere of the weighted sum of a random variables sequence of the class Pareto in the sub-linear expectation space. A random variable X is said to obey the Pareto distribution if the tail capacity is

(1.3)

The symbol “~” denotes equivalence, i.e., a_x\sim b_x denotes\underset{x\to \mathrm{\infty }}{lim}{a}_x/b_x=1.

2 Preliminaries

We use the framework and notations of Peng [10-12]. Let \left(\mathrm{\Omega },\mathcal{F}\right) be a given measurable space and let \mathcal{H} be a linear space of real functions defined on \left(\mathrm{\Omega },\mathcal{F}\right), if X_1,\cdots ,X_n\in \mathcal{H} then \phi \left(X_1,\cdots ,X_n\right)\in \mathcal{H} for each\mathrm{\forall }\phi \in C_{l,Lip}\left({\mathbb{R}}_n\right), where C_{l,Lip}\left({\mathbb{R}}_n\right) denotes the linear space of (local Lipschitz) functions \phi satisfying

\left|\phi \left(\mathit{x}\right)-\phi \left(\mathit{y}\right)\right|⩽c\left(1+{\left|\mathit{x}\right|}^m+{\left|\mathit{y}\right|}^m\right)\left|\mathit{x}-\mathit{y}\right|\text{,} \mathrm{\forall } \mathit{x}\text{,} \mathit{y}\in {\mathbb{R}}_n.

For some c>0, m\in \mathbb{N} depending on \phi, \mathcal{H} is considered as a space of random variables. In this case we denote X\in \mathcal{H}.

Definition 2.1 [11]　A sub-linear expectation \hat{E}:\mathcal{H} \to \mathbb{R}, \hat{E} satisfying the following properties:

1．Monotonicity: IfX⩾Y, then\hat{E}\left(X\right)⩾\hat{E}\left(Y\right);

2．Constant preserving: \hat{E}\left(c\right)=c;

3．Sub-additivity: \hat{E}\left(X\text{+}Y\right)⩽\hat{E}\left(X\right)+\hat{E}\left(Y\right);

4．Positive homogeneity: \hat{E}\left(\lambda X\right)=\lambda \hat{E}\left(X\right), for\lambda >0.

The triple(\mathrm{\Omega },\mathcal{H},\hat{E}) is called a sub-linear expectation space. \hat{E} is the sub-linear expectation. Let us denote the conjugate expectation \hat{\epsilon } of \hat{E} by

\hat{\epsilon }(X)= -\hat{E}(-X),\mathrm{\forall }X\in \mathcal{H}.

It is easily shown that

\begin{array}{l}\hat{\epsilon }(X)⩽\hat{E}(X),\hat{E}(X+c)=\hat{E}(X)+c,\\ |\hat{E}(X-Y)|⩽\hat{E}|X-Y|,\hat{E}(X-Y)⩾\hat{E}(X)-\hat{E}(Y),\mathrm{\forall }X,Y\in \mathcal{H}.\end{array}

If \hat{E}\left(Y\right)\text{=}\hat{\epsilon }\left(Y\right), then

\hat{E}\left(X+aY\right)= \hat{E}\left(X\right)+a\hat{E}\left(Y\right),\mathrm{\forall }a\in \mathbb{R}.

Definition 2.2 [11]　Let \mathcal{G}\subset \mathcal{F}, a function V:\mathcal{G}\to [0,1 ] is called a capacity if V satisfying the following properties:

1. V \left(\varnothing \right)=0,V \left(\mathrm{\Omega }\right)=1;

2. V \left(A\right)⩽ V\left(B\right),\mathrm{\forall }A\subseteq B,A,B \in \mathcal{G}.

It is sub-additive if V\left(A\cup B\right)⩽V\left(A\right)+V\left(B\right) for all A,B\in \mathcal{G}.

In the sub-linear space (\mathrm{\Omega },\mathcal{H},\hat{E}), we denote a pair (\mathbb{V};v ) of capacities by

\mathbb{V}(A)=inf\{\hat{E}\xi ,I(A )⩽\xi ,\xi \in \mathcal{H} \},v\left(A\right):=1-\mathbb{V}\left(A^c\right),\mathrm{\forall }A\in \mathcal{F},

where I(·) is the characteristic function, A^c is the complement set of A. From the definition,

v\left(A\right)⩽\mathbb{V}\left(A\right), \mathrm{\forall }A\in \mathcal{F}.

If I\left(A\right) \in \mathcal{H}, then

\mathbb{V}\left(A\right)=\hat{E}\left(I\left(A\right)\right),v \left(A\right)\text{=}\hat{\epsilon }\left(I\left(A\right)\right) .

If f⩽I( A)⩽g ,f,g\in \mathcal{H},

\hat{E}f⩽ \mathbb{V}(A)⩽\hat{E}g,\hat{\epsilon }f⩽ v(A)⩽\hat{\epsilon }g.

Therefore, Markov’s inequality

(2.1)\mathbb{V}\left(\left|X\right|⩾x\right)⩽\frac{\hat{E}\left({\left|X\right|}^p\right)}{x^p},\mathrm{\forall }X\in \mathcal{H},x>0,p>0

can be derived from I\left(\left|X\right|⩾x\right)⩽\frac{{\left|X\right|}^p}{x^p}\in \mathcal{H}.

Definition 2.3 [11]　We define the Choquet integrals/expecations

(2.2)C_V\left(X\right):={\int }_0^{\mathrm{\infty }} V\left(X>x\right)\text{d}x+ {\int }_{-\mathrm{\infty }}^0\left(V\left(X>x\right)-1\right)\text{d}x,

where the capacity V can be replaced by the upper capacity \mathbb{V} and the lower capacity \nu .

Definition 2.4 [11]　1. The countably sub-additive of \hat{E}: ifX⩽ \sum _{n=1}^{\mathrm{\infty }} X_n,where X,X_n\in \mathcal{H}, X⩾0, X_n⩾0, n⩾1, then \hat{E}\left(X\right) ⩽\sum _{n=1}^{\mathrm{\infty }}\hat{E}\left(X_n\right).

2. If V satisfying

V\left(\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\right)⩽\sum _{n=1}^{\mathrm{\infty }}V\left(A_n\right),\mathrm{\forall }{A}_n\in \mathcal{F},

then we call V countably sub-additive.

Definition 2.5 [11]　1. Identical distribution: Let ({\mathrm{\Omega }}_1,{\mathcal{H}}_1,{\hat{E}}_1) and ({\mathrm{\Omega }}_2,{\mathcal{H}}_2,{\hat{E}}_2) be two sub-linear expectation spaces and {\mathit{X}}_i\in\mathcal{H}, i=1,2. {\mathit{X}}_1 and {\mathit{X}}_2 are called identical distribution, which is denoted by {\mathit{X}}_1\stackrel{d}{=}{\mathit{X}}_2, if

{\hat{E}}_1\left[\phi \left({\mathit{X}}_1\right)\right]={\hat{E}}_2\left[\phi \left({\mathit{X}}_2\right)\right]\text{,}\mathrm{\forall }\phi \in {\text{C}}_{l,Lip}\left({\mathbb{R}}_n\right).

2. Independence: Let (\mathrm{\Omega },\mathcal{H},\hat{E}) be a sub-linear expectation space, and \mathit{Y}=\left(Y_1,\cdots ,Y_n\right)\text{,}{Y}_i\in \mathcal{H}, \mathit{X}=\left(X_1,\cdots ,X_m\right),X_i\in \mathcal{H}. \mathit{Y} is said to be independent to \mathit{X} under \hat{E}, if for each test function \phi \in C_{l,Lip}\left({\mathbb{R}}_m\times {\mathbb{R}}_n\right), we have \hat{E}\left(\phi \left(\mathit{X}\text{,}\mathit{Y}\right)\right)=\hat{E}\left(\hat{E}\left(\phi \left(\mathit{x}\text{,}\mathit{Y}\right)\right){|}_{x=X}\right), whenever for all \mathbf{x} and .

Definition 2.6 [14]　A random variable sequence \left\{X_n,n⩾1\right\} is said to converge to X almost surely V(\text{a}\text{.s} \text{.}V), denoted by X_n\to X\text{a}\text{.s} \text{.}V, if V\left(X_n\nrightarrow X\right)=0 as n\to \mathrm{\infty }. V can be replaced by \mathbb{V} and \nu respectively.

For A\in \mathcal{F}, \epsilon >0, we have

\begin{array}{rl}{X}_n\to X\text{a}\text{.s}\text{.}\mathbb{V}& \iff v\left(X_n\to X\right)=1\\ & \iff \mathbb{V}\left(\left|X_n-X\right|>\epsilon ,\text{i}\text{.o}.\right)=0.\end{array}

Proof　Since v\left(A\right)\text{+}\mathbb{V}\left(A^c\right)=1, we know that \mathbb{V}\left(X_n\nrightarrow X\right)=0\iff v(X_n\to X)=1. To prove that v \left(X_n\to X\right)=1\iff \mathbb{V}\left(\left|X_n-X\right|>\epsilon ,\text{i}\text{.o}.\right)=0, first of all, give

According to v\left(A\right)\text{+}\mathbb{V}\left(A^c\right)=1, we have

Then v\left(X_n\to X\right)=1\iff \mathbb{V}\left(\left|X_n-X\right|>\epsilon ,\text{i}\text{.o}.\right)=0.

Definition of convergence in capacity: \mathrm{\forall }\epsilon >0, if

\underset{n\to \mathrm{\infty }}{lim}V\left(\left|X_n-X\right|⩾\epsilon \right)=0,

then a random variable sequence \left\{X_n,n⩾1\right\} is said to converge to X in capacity V, denoted by X_n\stackrel{V}{\to }X. X_n\stackrel{V}{\to }{X}^{\text{+}}stands for \underset{n\to \mathrm{\infty }}{lim}V\left(X_n-X⩾\epsilon \right)=0. V can be replaced by \mathbb{V} and \nu respectively.

Later in this paper, \log x=\ln \left(max\left(x,\mathrm{e}\right)\right). The symbol “c” denotes a constant independent of n, which can take different values in different places. “\to” denotes the limit when n\to \mathrm{\infty }. a_x\sim b_x denotes \underset{x\to \mathrm{\infty }}{lim}{a}_x/b_x=1. “a_n\ll b_n” indicates the existence of a constant c>0 such that a_n⩽c{b}_n holds for sufficiently large n.

3 Main results

Theorem 3.1　 Let \left\{X_n,n⩾1\right\} be a sequence of non-negative independent random variables satisfying (1.1) and (1.2), and both the \hat{E} and \mathbb{V} are countably sub-additive. Note that

(3.1)X_{nj}=X_{j}I\left(X_j⩽b_n{c}_j\right)+b_n{c}_{j}I\left(X_j>b_n{c}_j\right), \mathrm{\forall }1⩽j⩽n.

Let b_n= C_n\log C_n. Then

(3.2)\frac{1}{b_n}{\sum }_{j=1}^n c_j^{-1}\left(X_j-\hat{E}{X}_{nj}\right)\stackrel{\mathbb{V}}{\to }{0}^{+}\text{.}

Corollary 3.1　 If \left\{X_n,n⩾1\right\} satisfies the condition of Theorem 3.1, and c_n=n, n⩾1, then

\frac{1}{\log n\log \log n} \sum _{j=1}^n{j}^{-1}\left(X_j-\hat{E}{X}_{nj}\right)\stackrel{\mathbb{V}}{\to } 0^{+}.

Theorem 3.2　 Let \left\{X_n,n⩾1\right\} be a sequence of independent and identically distributed, and both the \hat{E} and \mathbb{V} be countably sub-additive. If X_1 satisfies (1.3), then for \beta >0, we have

(3.3)\underset{n\to \mathrm{\infty }}{limsup}\frac{1}{{\log }^{\beta }n}\sum _{j=1}^n\frac{{\log }^{\beta -2}j}{j}{X}_j⩽\frac{1}{\beta }a.s \text{.}\mathbb{V}

and

(3.4)\underset{n\to \mathrm{\infty }}{lim inf}\frac{1}{{\log }^{\beta }n}\sum _{j=1}^n\frac{{\log }^{\beta -2}j}{j}{X}_j⩾ -\frac{1}{\beta } a.s.\mathbb{V}.

Note　Based on the Pareto distribution, we obtain the WLLN and the SLLN for weighted sums of independent random variables sequences. Theorem 3.1 and Corollary 3.1 generalize the WLLN results of Yang et al. [15] and Alder [2] in probability space to obtain the WLLN of a sequence of independent random variables in sub-linear expectation space, respectively. Theorem 3.2 generalizes the SLLN of Alder [1] in probability space to the sub-linear expectation space to obtain the SLLN of a sequence of independent identically distributed random variables in the sub-linear expectation space.

4 Proof of main result

Since \hat{E} is defined only for X\in \mathcal{H}, an extension E^{\text{*}}: E^{\ast }( X)=inf\{\hat{E}[Y] :X⩽Y, Y\in \mathcal{H}\} can be defined in the space of random variables such that E^{\text{*}} is defined for all X.

Lemma 4.1 [16]　 E^{\ast }( X) is a sub-linear expectation in the space of random variables and has the following properties:

E^{\ast }[ X]=\hat{E}[X ],\mathrm{\forall }X\in \mathcal{H};\mathbb{V}(A)= E^{\ast }\left[I_A\right],\mathrm{\forall }A\in \mathcal{F}.

Lemma 4.2 [16, 17]　 For k=1,\dots ,n -1, if X_k and \left(X_{k+1},\dots ,X_n\right) are independent of each other and satisfy \hat{E}{X}_k⩽0, k=1,\dots , n, S_k=\sum _{i=1}^k{X}_i, then we have

(4.1)\hat{E}\left({\left|\underset{k⩽n}{max}{S}_k\right|}^p\right)⩽2^{2-p}\sum _{k=1}^n\hat{E}{X}_k^2,\text{1}⩽p⩽ 2,

(4.2)\mathbb{V}\left(S_n\ge x\right)\le c\frac{{\displaystyle \sum _{k=1}^n\hat{E}{X}_k^2}}{x^2},\mathrm{\forall }x> 0.

Lemma 4.3 [17] (Borel-Cantelli lemma)　Let \left\{A_n,n⩾1\right\} be a column of events in \mathcal{F}. Assume that V is a capacity with countably sub-additive. If , then

V\left(A_n\text{; i}\text{.o}\text{.}\right)=0 ,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\left\{A_n;\text{i}\text{.o}\text{.}\right\}=\bigcap _{n=1}^{\mathrm{\infty }}\bigcup _{i=n}^{\mathrm{\infty }}{A}_i.

Lemma 4.4 [16]　 If \hat{E} is countably sub-additive, then \hat{E}\left(\left|X\right|\right)⩽C_{\mathbb{V}}\left(\left|X\right|\right) , \mathrm{\forall }X\in \mathcal{H}.

Lemma 4.5 [16] (Hölder inequality)　 Let p, q>1 be two real numbers satisfying p^{-1}+q^{-1}\text{=}1. Then for two random variables X,Y\in \mathcal{H}, we have

\hat{E}\left(\left|XY\right|\right)⩽{\left(\hat{E}{\left|X\right|}^p\right)}^{1/p}{\left(\hat{E}{\left|Y\right|}^q\right)}^{1/q}.

Proof of Theorem 3.1 \frac{1}{b_n}{\sum }_{j=1}^n{c}_j^{-1}(X_j-\hat{E}{X}_{nj}) can be decomposed into

\begin{array}{rl}\frac{1}{b_n}{\displaystyle {\sum }_{j=1}^n{c}_j^{-1}\left(X_j-\hat{E}{X}_{nj}\right)}& =\frac{1}{b_n}{\displaystyle \sum _{j=1}^n{c}_j^{-1}\left(X_j-X_{nj}\right)}\text{+}\frac{1}{b_n}{\displaystyle \sum _{j=1}^n{c}_j^{-1}\left(X_{nj}-\hat{E}{X}_{nj}\right)}\\ & :=I_1+I_2.\end{array}

To prove that (3.2) holds, it is only necessary to prove that

(4.3)I_1=\frac{1}{b_n} \sum _{j=1}^n{c}_j^{-1}\left(X_j-X_{nj}\right)\stackrel{\mathbb{V}}{\to } 0

and

(4.4)I_2=\frac{1}{b_n} \sum _{j=1}^n{c}_j^{-1}\left(X_{nj}-\hat{E}{X}_{nj}\right)\stackrel{\mathbb{V}}{\to }{0}^{\text{+}} .

First, from (1.1) and (1.2), we have

\begin{array}{rl}\mathbb{V}\left(\frac{1}{b_n}\left|\sum _{j=1}^n{c}_j^{-1}\left(X_j-X_{nj}\right)\right|>\epsilon \right)& ⩽\mathbb{V}\left(\bigcup _{j=1}^n\left(X_j\ne X_{nj}\right)\right)⩽\sum _{j=1}^n\mathbb{V}\left(X_j>b_n{c}_j\right)\\ & =\sum _{j=1}^n\frac{1}{b_n{c}_j+c_j}=\frac{1}{b_n+1}\sum _{j=1}^n\frac{1}{c_j}\\ & =\frac{C_n}{1+C_n\log C_n}\to 0,\mathrm{\forall }\epsilon >0.\end{array}

Therefore (4.3) holds.

Next, we prove (4.4). Lemma 4.5 shows that (\hat{E}{X}_j{)}^2⩽ \hat{E}{X}_j^2, so we have

(4.5)\hat{E}(X_j-\hat{E}{X}_j{)}^2⩽ 2\hat{E}(X_j^2\text{+}{(\hat{E}{X}_j)}^2)\ll \hat{E}{X}_j^2.

From (1.1), we have

(4.6)\begin{array}{rl}{c}_j^{-2}{\int }_0^{b_n^2{c}_j^2}\mathbb{V}\left(X_j^2>t\right)\text{d}t& =c_j^{-2}{\int }_0^{b_n^2{c}_j^2}\frac{1}{t^{1/2}+c_j}\text{d}t\text{( let}{t}^{1/2}=x)\\ & =c_j^{-2}{\int }_0^{b_n{c}_j}\frac{2x}{x+c_j}\text{d}x\\ & \ll c_j^{-1}\left(b_n-\ln \left(b_n+1\right)\right).\end{array}

Since the \hat{E} is countably sub-additive, it follows from Lemma 4.4 that \hat{E}\left|X_j\right|⩽C_{\mathbb{V}}\left(\left|X_j\right|\right). Then by X_j\in \mathcal{H} and Lemma 4.1, we have \hat{E}\left(X_j\right)=E^{\ast }\left(X_j\right). Since \{c_j^{-1}(X_{nj}-\hat{E}{X}_{nj}),1⩽j⩽n\}\in \mathcal{H} is an independent sequence with zero mean, we can apply (4.2) of Lemma 4.2 to c_j^{-1}(X_{nj}-\hat{E}{X}_{nj}). Then by (1.2), (2.2), (3.1), (4.5) and (4.6), we have

\begin{array}{rl}& \mathbb{V}\left(\frac{1}{b_n}\sum _{j=1}^n{c}_j^{-1}\left(X_{nj}-\hat{E}{X}_{nj}\right)⩾\epsilon \right)\\ & \ll \frac{1}{b_n^2}\sum _{j=1}^n{c}_j^{-2}\hat{E}{\left(X_{nj}-\hat{E}{X}_{nj}\right)}^2\\ & \ll \frac{1}{b_n^2}\sum _{j=1}^n{c}_j^{-2}\hat{E}{X}_{nj}^2\\ & =\frac{1}{b_n^2}\sum _{j=1}^n{c}_j^{-2}{E}^{\ast }{X}_{nj}^2\\ & =\frac{1}{b_n^2}\sum _{j=1}^n{c}_j^{-2}\left(E^{\ast }{X}_j^{2}I\left(X_j⩽b_n{c}_j\right)+b_n^2{c}_j^2\mathbb{V}\left(X_j>b_n{c}_j\right)\right)\\ & =\frac{1}{b_n^2}\sum _{j=1}^n\left(c_j^{-2}{E}^{\ast }{X}_j^{2}I\left(X_j⩽b_n{c}_j\right)+b_n^2\mathbb{V}\left(X_j>b_n{c}_j\right)\right)\\ & ⩽\frac{1}{b_n^2}\sum _{j=1}^n\left(c_j^{-2}{C}_{\mathbb{V}}\left(X_j^{2}I\left(X_j⩽b_n{c}_j\right)\right)+b_n^2\mathbb{V}\left(X_j>b_n{c}_j\right)\right)\\ & \ll \frac{1}{b_n^2}\sum _{j=1}^n\left(c_j^{-2}{\int }_0^{b_n^2{c}_j^2}\mathbb{V}\left(X_j^2>t\right)\text{d}t+\frac{b_n^2}{b_n{c}_j+c_j}\right)\\ & \ll \sum _{j=1}^n\left(\frac{1}{b_n^2}\left(\frac{1}{c_j}\left(b_n-\ln \left(b_n+1\right)\right)+\frac{1}{c_j\left(b_n+1\right)}{b}_n^2\right)\right)\\ & \ll \frac{C_n}{b_n}\to 0.\end{array}

Therefore (4.4) holds. Theorem 3.1 is proved.

Proof of Corollary 3.1　When c_n=n, C_n\text{=} \sum _{j=1}^n\frac{1}{c_j}=\sum _{j=1}^n\frac{1}{j}\ll \log n\to \mathrm{\infty }, so c_n=n also satisfies the conditions of Theorem 3.1. According to b_n= C_n\log C_n, we know that b_n=\log n\log \log n, so we have

\frac{1}{\log n\log \log n} {\sum }_{j=1}^n{j}^{-1}\left(X_j-\hat{E}{X}_{nj}\right)\stackrel{\mathbb{V}}{\to } {\text{0}}^{\text{+}}.

Proof of Theorem 3.2　 \mathrm{\forall }j⩾1, let a_j=\frac{{\log }^{\beta -2}j}{j}, b_j={\log }^{\beta }j, c_j=j{\log }^{\alpha }j for . Let

(4.7)

As

(4.8)\begin{array}{rl}\frac{1}{{\log }^{\beta }n}\sum _{j=1}^n\frac{{\log }^{\beta -2}j}{j}{X}_j& =\frac{1}{b_n}\sum _{j=1}^n{a}_j{X}_j\\ & =\frac{1}{b_n}\sum _{j=1}^n{a}_j\left(X_j-{\tilde{X}}_j\right)\text{+}\frac{1}{b_n}\sum _{j=1}^n{a}_j\left({\tilde{X}}_j-\hat{E}{\tilde{X}}_j\right)\\ & +\frac{1}{b_n}\sum _{j=1}^n{a}_j\left(\hat{E}{\tilde{X}}_j\right)\\ & :=I_1+I_2+I_3.\end{array}

According to (4.8), to prove that (3.3) holds, we only need to prove that

(4.9)I_1=\frac{1}{b_n} \sum _{j=1}^n{a}_j\left(X_j-{\tilde{X}}_j\right)\to 0 \text{a}\text{.s}\text{.}\mathbb{V};

(4.10)\underset{n\to \mathrm{\infty }}{limsup}{I}_2\text{=}\underset{n\to \mathrm{\infty }}{limsup}\frac{1}{b_n} \sum _{j=1}^n{a}_j\left({\tilde{X}}_j-\hat{E}{\tilde{X}}_j\right)⩽0\text{a}\text{.s}\text{.}\mathbb{V}\text{;}

(4.11)\underset{n\to \mathrm{\infty }}{limsup}{I}_3= \underset{n\to \mathrm{\infty }}{limsup} \frac{1}{b_n}\sum _{j=1}^n{a}_j\left(\hat{E}{\tilde{X}}_j\right)⩽\frac{1}{\beta }.

Taking , let the even function g\left(x\right)\in C_{l,Lip}\left(\mathbb{R}\right), satisfying 0⩽g\left(x\right)⩽1 for all x. When \left|x\right|⩽u, g\left(x\right)=1; when \left|x\right|>1, g\left(x\right)=0, the following holds:

I\left(\left|x\right|⩽u\right)⩽g\left(\left|x\right|\right)⩽I\left(\left|x\right|⩽1\right),I \left(\left|x\right|>1\right)⩽1 -g\left(\left|x\right|\right)⩽I\left(\left|x\right|>u\right).

Therefore, we have

(4.12)\begin{array}{rl}\sum _{j=1}^{\mathrm{\infty }}\mathbb{V}\left(\left|X_j\right|⩾c_j\right)& ⩽\sum _{j=1}^{\mathrm{\infty }}\hat{E}\left(1-g\left(\frac{\left|X_j\right|}{c_j}\right)\right)\\ & =\sum _{j=1}^{\mathrm{\infty }}\hat{E}\left(1-g\left(\frac{\left|X_1\right|}{c_j}\right)\right)\\ & ⩽\sum _{j=1}^{\mathrm{\infty }}\mathbb{V}\left(\left|X_1\right|>\mu c_j\right).\end{array}

From (1.3) and (4.12), we have

Combining (4.7), we have

Since \mathbb{V} is countably sub-additive, by Lemma 4.3, we have

\mathbb{V}\left(X_j\ne {\tilde{X}}_j\text{, i}\text{.o}\text{.}\right)=0.

Further according to \underset{n\to \mathrm{\infty }}{lim}\frac{1}{b_n}\text{=}\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{\log }^{\beta }n}\text{=}0, we have

\left|I_1\right|=\frac{1}{b_n} \left|{\displaystyle \sum _{j=1}^n{a}_j\left(X_j-{\tilde{X}}_j\right)}\right|\le \frac{1}{b_n} {\displaystyle \sum _{j=1}^n\left|a_j\left(X_j-{\tilde{X}}_j\right)\right|} \to 0\text{a} \text{.s}\text{.}\mathbb{V},

Therefore (4.9) holds.

Next, prove (4.10). First, for every n, there exists k such that . For sufficiently large n, we have \frac{b_{2^k}}{b_{2^{k-1}}}\text{=}\frac{{\log }^{\beta }{2}^k}{{\log }^{\beta }{2}^{k-1}}\to 1. Further, we have

(4.13)

From (4.12), to prove (4.10), it is only necessary to prove that

(4.14)

Next, since \mathbb{V} is countably sub-additive, applying Lemma 4.3, it is sufficient to prove that

(4.15)

Therefore, (4.14) holds.

First, since the \hat{E} is countably sub-additive, by Lemma 4.4, we have \hat{E}\left|X_j\right|⩽C_{\mathbb{V}}\left(\left|X_j\right|\right). From X_j\in \mathcal{H} and Lemma 4.1, we have \hat{E}\left(X_j\right)=E^{\ast }\left(X_j\right). Further, the following equations can be derived from (1.3), (2.2) and (4.12):

(4.16)\begin{array}{rl}\hat{E}{\tilde{X}}_j^2& =\hat{E}\left(c_j^{2}I\left(\left|X_j\right|>c_j\right)+X_j^{2}I\left(\left|X_j\right|⩽c_j\right)\right)\\ & =E^{\ast }\left(c_j^{2}I\left(\left|X_j\right|>c_j\right)+X_j^{2}I\left(\left|X_j\right|⩽c_j\right)\right)\\ & ⩽c_j^2\mathbb{V}\left(\left|X_j\right|>c_j\right)+E^{\ast }{X}_j^{2}I\left(\left|X_j\right|⩽c_j\right)\\ & ⩽c_j^2\mathbb{V}\left(\left|X_1\right|>\mu c_j\right)+E^{\ast }{X}_1^{2}I\left(\left|X_1\right|⩽c_j\right)\\ & ⩽c_j^2\mathbb{V}\left(\left|X_1\right|>\mu c_j\right)+C_{\mathbb{V}}\left(X_1^{2}I\left(\left|X_1\right|⩽c_j\right)\right)\\ & \ll c_j+{\int }_1^{c_j^2}\mathbb{V}\left(X_1>t^{1/2}\right)\text{d}t\\ & \ll c_j+{\int }_1^{c_j^2}\frac{1}{t^{1/2}}\text{d}t\\ & \ll c_j.\end{array}

Therefore, from (4.16), we have

(4.17)\begin{array}{rl}\sum _{j=1}^{2^n}{a}_j^2\hat{E}{\tilde{X}}_j^2& ⩽\sum _{j=1}^{2^n}{a}_j^2{c}_j\\ & =\sum _{j=1}^{2^n}\frac{{\log }^{2\beta -4}j}{j^2}\cdot j{\log }^{\alpha }j\\ & =\sum _{j=1}^{2^n}\frac{{\log }^{2\beta \text{+}\alpha -4}j}{j}\\ & \ll {\log }^{2\beta \text{+}\alpha -3}{2}^n\\ & \ll n^{2\beta \text{+}\alpha -3}.\end{array}

From Definition 2.5, \{{\tilde{X}}_j-\hat{E}{\tilde{X}}_j,j⩾1\}\in \mathcal{H} and \{\frac{1}{c_j}\left({\tilde{X}}_j-\hat{E}{\tilde{X}}_j\right),j⩾1 \}\in \mathcal{H} are also a sequence of independent random variables with zero mean. The following equation can be deduced from (2.1), (4.1), (4.5) and (4.17):

(4.18)

From (4.18), (4.15) holds when 3-\alpha >1. Since , (4.15) holds. Then, according to Lemma 4.3, (4.14) holds. Further, (4.10) also holds according to (4.13).

Next, prove (4.11), i.e., prove that

According to (1.1) and (4.12), we have

Therefore, it can be proved that

(4.19)H\to 0.

Furthermore, since \hat{E} is countably sub-additive, we have \hat{E}\left|X_j\right|⩽C_{\mathbb{V}}\left(\left|X_j\right|\right) by Lemma 4.4. Then, since X_j\in \mathcal{H} and Lemma 4.1, we have \hat{E}\left(X_j\right)=E^{\ast }\left(X_j\right). Combining with (1.3) and (2.2), we have

\begin{array}{rl}\left|E^{\text{*}}{X}_{n}I\left(\left|X_n\right|⩽c_n\right)\right|& ⩽E^{\text{*}}\left|X_n\right|I\left(\left|X_n\right|⩽c_n\right)\\ & =E^{\text{*}}\left|X_1\right|I\left(\left|X_1\right|⩽c_n\right)⩽C_{\mathbb{V}}\left|X_1\right|I\left(\left|X_1\right|⩽c_n\right)\\ & ={\int }_0^{\mathrm{\infty }}\mathbb{V}\left(\left|X_1\right|>t\right)\text{d}t\sim {\int }_1^{c_n}\mathbb{V}\left(\left|X_1\right|>t\right)\text{d}t\\ & \sim {\int }_1^{c_n}\frac{1}{t}\text{d}t\text{=}\log c_n\sim \log n.\end{array}

Thus,

(4.20)\sum _{j=1}^n\frac{a_j}{b_n}\left|E^{\text{*}}{X}_{n}I\left(\left|X_n\right|⩽c_n\right)\right|⩽\sum _{j=1}^n\frac{a_j}{b_n} \log j=\frac{1}{{\log }^{\beta }n}\sum _{j=1}^n\frac{{\log }^{\beta -1}j}{j}\to \frac{1}{\beta }.

Combining (4.19) and (4.20), we have

\underset{n\to \mathrm{\infty }}{limsup}{I}_3⩽ \frac{1}{\beta }.

In summary, (4.9)‒(4.11) all hold, so (3.3) holds.

Obviously, \left\{-X_j\text{;}j⩾1\right\} also satisfies the condition of Theorem 3.2. By replacing \left\{X_j;j⩾1\right\} in (3.3) with \left\{-X_j;j⩾1\right\}, we can get (3.4) also holds. Theorem 3.2 is proved.

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