Convergence rates in the law of large numbers under sublinear expectations

TL;DR

This paper investigates how quickly averages of i.i.d. random variables converge under sublinear expectations, establishing strong convergence results in $L^p$ and quasi sure senses.

Contribution

It introduces new convergence rate results for the law of large numbers within the framework of sublinear expectations, extending classical results.

Findings

Established strong $L^p$-convergence for i.i.d. variables under sublinear expectations.

Proved strongly quasi sure convergence versions of the law of large numbers.

Extended classical convergence results to the sublinear expectation setting.

Abstract

In this note, we study convergence rates in the law of large numbers for independent and identically distributed random variables under sublinear expectations. We obtain a strong $L^p$-convergence version and a strongly quasi sure convergence version of the law of large numbers.

Equations

$$\sum _{n=1}^{\infty }P(|Y_n-Y|>\epsilon )<\infty ,$$

$$\sum _{n=1}^{\infty }{n}^{\frac{\alpha }{p}-\frac{1}{p}-2}E\left[{\left(|S_n|-\epsilon n^{\frac{1}{p}}\right)}^{+}\right]<\infty ,\forall \epsilon >0,$$

$$\displaystyle\begin{array}[]{ccccccc}Y_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}Y&\Rightarrow&Y_{n}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}Y&\Rightarrow&Y_{n}\stackrel{{\scriptstyle P}}{{\longrightarrow}}Y&\Rightarrow&Y_{n}\stackrel{{\scriptstyle d}}{{\longrightarrow}}Y\\ \Uparrow&&&&\Uparrow&&\\ Y_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}Y&\Rightarrow&Y_{n}\stackrel{{\scriptstyle L^{p}}}{{\longrightarrow}}Y&\Rightarrow&Y_{n}\stackrel{{\scriptstyle L^{1}}}{{\longrightarrow}}Y&&\end{array}$$

$$|\phi (x)-\phi (y)|\le C(1+|x{|}^m+|y{|}^m)|x-y|,\forall x,y\in {\mathbb{R}}^n$$

$$\hat{E}[X]=\underset{P\in \mathcal{P}}{\sup }{E}_P[X],\forall X\in \mathcal{H}.$$

$$V(A):=\hat{E}[I_A],v(A):=-\hat{E}[-I_A],\forall A\in \mathcal{B}(\Omega ).$$

$$V(\varnothing )=0,V(\Omega )=1,V(A)+v(A^c)=1,$$

$${\Vert \cdot \Vert }_p:X\in \mathcal{H}\mapsto (\hat{E}[|X{|}^p]{)}^{\frac{1}{p}}$$

$${\hat{E}}_1[\phi (X_1)]={\hat{E}}_2[\phi (X_2)],\forall \phi \in C_{l,lip}({\mathbb{R}}^n).$$

$$\hat{E}[\phi (X,Y)]=\hat{E}[\hat{E}[\phi (x,Y){]}_{x=X}],$$

$$aX+b\bar{X}\stackrel{d}{=}\sqrt{a^2+b^2}X\text{mbox}foranya,b\ge 0,$$

$$\displaystyle\begin{array}[]{ccccccccc}&&&&X_{n}\stackrel{{\scriptstyle S_{p}\mbox{-}q.s.}}{{\longrightarrow}}X&\Rightarrow&X_{n}\stackrel{{\scriptstyle q.s}}{{\longrightarrow}}X&&\\ &&&&\Uparrow&&\Uparrow&&\\ X_{n}\stackrel{{\scriptstyle V}}{{\longrightarrow}}X&\Leftarrow&X_{n}\stackrel{{\scriptstyle L^{p}}}{{\longrightarrow}}X&\Leftarrow&X_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}X&\Rightarrow&X_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}X&\Rightarrow&X_{n}\stackrel{{\scriptstyle V}}{{\longrightarrow}}X\\ \end{array}$$

$$X_n\stackrel{q.s.}{⟶}X\Rightarrow X_n\stackrel{V}{⟶}X\Rightarrow X_n\stackrel{d}{⟶}X.$$

$$\underset{n\to \infty }{\lim }\hat{E}[\phi (\overline{S_n})]=\hat{E}[\phi (\xi )],$$

$$\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{n}\right|}^p\right]=\frac{1}{n^{p/2}}\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{\sqrt{n}}\right|}^p\right].$$

$$\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{\sqrt{n}}\right|}^p\right]\to \hat{E}[|\xi {|}^p],\text{mbox}asn\to \infty ,$$

$$\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{\sqrt{n}}\right|}^p\right]\ge \frac{c_p}{2},\forall n\ge N.$$

$$\sum _{n=1}^{\infty }\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{n}\right|}^p\right]\ge \sum _{n=N}^{\infty }\frac{1}{n^{p/2}}\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{\sqrt{n}}\right|}^p\right]\ge \frac{c_p}{2}\sum _{n=N}^{\infty }\frac{1}{n^{p/2}}=\infty .$$

$$\hat{E}\left[{\left|{\tilde{S}}_n\right|}^{\alpha }\right]$$

$$\sum _{n=1}^{\infty }\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{n}\right|}^{\alpha }\right]\le C_{\alpha }\sum _{n=1}^{\infty }\frac{\hat{E}[|{\tilde{X}}_1{|}^{\alpha }]}{n^{\frac{\alpha }{2}}}<\infty .$$

$$\sum _{n=1}^{\infty }\hat{E}\left[{\left|\frac{{\tilde{S}}_n}{n}\right|}^p\right]$$

$$\sum _{n=1}^{\infty }{n}^{-2}E[|S_n{|}^{\alpha }]<\infty .$$

$$\sum _{n=1}^{\infty }{n}^{-\beta }\hat{E}[|S_n{|}^p]<\infty .$$

$$\frac{S_n}{n}\stackrel{S-L^p}{⟶}\mu \iff p>2.$$

$$\frac{{\tilde{S}}_n}{n}\stackrel{S_{\beta }\text{mbox}-q.s.}{⟶}0,$$

$$\sum _{k=1}^{\infty }\hat{E}\left[{\left|\frac{S_n}{n}-\mu \right|}^{\beta }\right]<\infty ,$$

$$\hat{E}\left[\sum _{k=1}^{\infty }{\left|\frac{S_n}{n}-\mu \right|}^{\beta }\right]$$

$$\sum _{k=1}^{\infty }{\left|\frac{S_n}{n}-\mu \right|}^{\beta }<\infty ,q.s.\text{mbox}forany2<\beta \le \alpha .$$

$$\left|\frac{S_n}{n}-\mu \right|<1.$$

$$\sum _{n=M(\omega )}^{\infty }{\left|\frac{S_n}{n}-\mu \right|}^{\beta }\le \sum _{n=M(\omega )}^{\infty }{\left|\frac{S_n}{n}-\mu \right|}^{\alpha },$$

Full text

Convergence rates in the law of large numbers under sublinear expectations

Ze-Chun Hu, Ning-Hua Liu, Ting Ma

College of Mathematics, Sichuan University, China Corresponding author: College of Mathematics, Sichuan University, Chengdu 610065, ChinaE-mail address: zchu@scu.edu.cn (Z.-C. Hu), 2639428701@qq.com (N.-H. Liu), matingting2008@scu.edu.cn (T. Ma)

Abstract  In this note, we study convergence rates in the law of large numbers for independent and identically distributed random variables under sublinear expectations. We obtain a strong $L^{p}$-convergence version and a strongly quasi sure convergence version of the law of large numbers.

Key words  Law of large number, sublinear expectation, convergence rate.

Mathematics Subject Classification (2010)  60F15, 60F25

1 Introduction

Let $\{X,X_{n},n\geq 1\}$ be a sequence of independent and identically distributed (i.i.d.) random variables in a probability space $(\Omega,\mathcal{F},P)$. Define $S_{n}=X_{1}+\cdots+X_{n}$, $n\in\mathbb{N}$. If $E|X|<\infty$, then by the law of large numbers, we know that $\frac{S_{n}}{n}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}\mu$, where $\mu=E[X]$. In fact, we also have $\frac{S_{n}}{n}\stackrel{{\scriptstyle L^{1}}}{{\longrightarrow}}\mu$ by the martingale theory.

Hsu and Robbins (1947) introduced a new kind of convergence named “complete convergence”. Let $\{Y,Y_{n},n\geq 1\}$ be a sequence of random variables. $\{Y_{n},n\geq 1\}$ is said to completely converge to $Y$, if for any $\varepsilon>0$,

[TABLE]

which is denoted by $Y_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}Y$. Obviously, $Y_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}Y\Rightarrow Y_{n}\stackrel{{\scriptstyle P}}{{\longrightarrow}}Y$. In fact, by the Borel-Cantelli lemma, we know that $Y_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}Y\Rightarrow Y_{n}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}Y$.

Hsu and Robbins (1947) proved that if $E[X^{2}]<\infty$ and $E[X]=\mu$, then $\frac{S_{n}}{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}\mu$. Erdös (1949) proved the converse result. Baum and Katz (1965) extended the Hsu-Robbins-Erdös theorem. Below is a special case of the Baum-Katz theorem.

Theorem 1.1

(Baum and Katz (1965)). Let $\alpha\geq 1$. Suppose that $\{X,X_{n},n\geq 1\}$ is a sequence of i.i.d. random variables with partial sum $S_{n}=\sum_{i=1}^{n}X_{i}$, $n\in\mathbb{N}$. Then the condition $E|X|^{\alpha}<\infty$ and $EX=0$ is equivalent to $\sum_{n=1}^{\infty}n^{\alpha-2}P\left(|S_{n}|>n\varepsilon\right)<\infty,\ \ \forall\varepsilon>0.$

Lanzinger (1998), Gut and Stadtmüller (2011), Chen and Sung (2014) extended the results of Baum and Katz (1965).

Chow (1988) first investigated the complete moment convergence and obtained the following result. Let $\alpha\geq 1$, $p\leq\alpha$ and $p<2$. Suppose that $\{X,X_{n},n\geq 1\}$ is a sequence of i.i.d. random variables with $E[X]=0$. If $E[|X|^{\alpha}+|X|\log^{+}|X|]<\infty$, then

[TABLE]

where $x^{+}=\max\{0,x\}$. Chow’s result has been generalized in various directions. Refer to Qiu and Chen (2014), Li and Hu (2017) and the references therein.

Li and Hu (2017) introduced a new convergence called “strong $L^{p}$-convergence”. Let $\{Y,Y_{n},n\geq 1\}$ be a sequence of random variables, and $p>0$. $Y_{n}$ is said to strongly $L^{p}$-converge to $Y$ if $\sum_{n=1}^{\infty}E[|Y_{n}-Y|^{p}]<\infty,$ which is denoted by $Y_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}Y$. Obviously, $Y_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}Y\Rightarrow Y_{n}\stackrel{{\scriptstyle L^{p}}}{{\longrightarrow}}Y$. By Markov’s inequality, $Y_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}Y\Rightarrow Y_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}Y$. Then for $p\geq 1$, we have the following diagram:

[TABLE]

In particular, for $p>1$, $Y_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}Y$ implies both $Y_{n}\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}Y$ and $Y_{n}\stackrel{{\scriptstyle L^{1}}}{{\longrightarrow}}Y$.

Recently, Hu and Sun (2018) studied convergence rates in the law of large numbers for i.i.d. random variables. They obtained a strong $L^{p}$-convergence version and a strong almost sure convergence version of the law of large numbers in a probability space.

The motivation of this note is to study convergence rates in the law of large numbers for i.i.d. random variables under sublinear expectations, and extend some results in a probability space to a sublinear expectation space.

Motivated by the risk measures, superhedge pricing and modeling uncertainty in finance, Peng (2006, 2007, 2008a, 2008b, 2009, 2010) initiated the notion of i.i.d. random variables under sublinear expectations, and proved the central limit theorems and the weak law of large numbers among others.

Hu and Zhou (2015) presented some multi-dimensional laws of large numbers under sublinear expectations without the requirement of identical distribution. Chen (2016) proved a strong law of large numbers (SLLNs) for i.i.d. random variables under capacities induced by sublinear expectations. Hu and Chen (2016) presented three laws of large numbers for independent random variables without the requirement of identical distribution. Zhang (2016) showed that Kolmogorov’s SLLNs holds for i.i.d. random variables under a continuous sublinear expectation if and only if the corresponding Choquet integral is finite. Chen et al. (2017) investigated some SLLNs for sublinear expectation without independence. Hu and Yang (2017) obtained a SLLNs for i.i.d. random variables under one-order type moment condition. Hu (2018) obtained a SLLNs for a sequence of independent random variables satisfying a controlled 1st moment condition under sublinear expectations. Chen et al. (2019) established a kind of SLLNs for capacities with a new notion of exponential independence for random variables under an upper expectation.

We refer to Marinacci (1999), Maccheroni and Marinacci (2005), Cozman (2010), Li and Chen (2011), Chen (2012), Chen et al. (2013), Agahi et al. (2013), Zhang and Chen (2015), Hu et al. (2016), Wu and Jiang (2018) for more results on SLLNs for capacity, nonlinear expectations or sublinear expectations. We also refer to Hu and Zhou (2019) and Zhang (2019) for some recent results on the convergence of random variables under sublinear expectations.

The rest of this note is organized as follows. In Section 2, we recall some basic notions and results on sublinear expectations. In Section 3, we present our main results and give the proofs. In the final section, we mention some questions.

2 Sublinear expectations

In this section, we introduce some basic definitions and notations about sublinear expectation. Refer to Peng (2010) for more details.

Let $\Omega$ be a given set and let $\mathcal{H}$ be a linear space of real functions defined on $\Omega$ such that for any constant number $c$, $c\in\mathcal{H}$; if $X\in\mathcal{H}$, then $|X|\in\mathcal{H}$; if $X_{1},\cdots,X_{n}\in\mathcal{H}$, then for any $\varphi\in C_{l,lip}(\mathbb{R}^{n})$, $\varphi(X_{1},\cdots,X_{n})\in\mathcal{H}$, where $C_{l,lip}(\mathbb{R}^{n})$ denotes the linear space of functions $\varphi$ satisfying

[TABLE]

for some $C>0$, $m\in\mathbb{N}$ depending on $\varphi$. For $n\in\mathbb{N}$, let $\mathcal{H}^{n}=\{X=(X_{1},\cdots,X_{n}),X_{i}\in\mathcal{H},\forall i=1,\cdots,n\}$ denote $n$-dimensional random vector space.

Definition 2.1

(Definition 1.1 of Peng (2010))* A sublinear expectation $\hat{E}$ on $\mathcal{H}$ is a functional $\hat{E}:\mathcal{H}\rightarrow\mathbb{R}$ satisfying the following properties $:\forall$ $X,Y\in\mathcal{H}$,*

(i) Monotonicity: $\hat{E}[X]\geq\hat{E}[Y],\ \mbox{if}\ X\geq Y$;

(ii) Constant preserving: $\hat{E}[c]=c,\ \forall\ c\in\mathbb{R}$;

(iii) Sub-additivity: $\hat{E}[X+Y]\leq\hat{E}[X]+\hat{E}[Y]$;

*(iv) Positive homogeneity: $\hat{E}[\lambda X]=\lambda\hat{E}[X],\ \forall\lambda\geq 0$.

The triple $(\Omega,\mathcal{H},\hat{E})$ is called a sublinear expectation space.*

In the following, we assume that $\Omega$ is a complete separable metric space and let $\mathcal{B}(\Omega)$ denote the Borel $\sigma$-algebra of $\Omega$. Further we assume that there exists a family $\mathcal{P}$ of probability measures on $(\Omega,\mathcal{B}(\Omega))$ such that

[TABLE]

Suppose that for any $A\in\mathcal{B}(\Omega)$, $I_{A}\in\mathcal{H}$. A pair of capacities associated with $\hat{E}[\cdot]$ are defined by

[TABLE]

It is easy to check that

[TABLE]

where $A^{c}$ is the complementary set of $A$, $A\in\mathcal{B}(\Omega)$. For $p\in[1,+\infty)$, the map

[TABLE]

forms a seminorm on $\mathcal{H}$.

Definition 2.2

(Definition I.3.1 of Peng (2010))* Let $(\Omega_{i},\mathcal{H}_{i},\hat{E}_{i}),i=1,2$ be two sublinear expectation spaces and $X_{i}\in\mathcal{H}_{i}^{n},i=1,2$. $X_{1}$ and $X_{2}$ are called identically distributed, which is denoted by $X_{1}\stackrel{{\scriptstyle d}}{{=}}X_{2}$, if*

[TABLE]

Definition 2.3

(Definition I.3.10 of Peng (2010))* Let $(\Omega,\mathcal{H},\hat{E})$ be a sublinear expectation space, and $X\in\mathcal{H}^{m}$, $Y\in\mathcal{H}^{n}$, $n,m\in\mathbb{N}$. $Y$ is said to be independent to $X$ under $\hat{E}[\cdot]$, if for each test function $\varphi\in C_{l,lip}(\mathbb{R}^{n+m})$, we have*

[TABLE]

whenever $\bar{\varphi}(x):=\hat{E}[|\varphi(x,Y)|]<\infty$ for all $x$ and $\hat{E}[|\bar{\varphi}(X)|]<\infty$.

Definition 2.4

(Proposition I.3.15 of Peng (2010))* A sequence of random variables $\{X_{n},n\geq 1\}$ on $(\Omega,\mathcal{H},\hat{E})$ is said to be independent and identically distributed, if $X_{i}\stackrel{{\scriptstyle d}}{{=}}X_{1}$ and $X_{i+1}$ is independent to $(X_{1},\cdots,X_{i})$ for each $i\geq 1$.*

Definition 2.5

(Definition II.1.4 of Peng (2010))* A $d$-dimensional random vector $X=(X_{1},\cdots,X_{d})^{T}$ on a sublinear expectation space $(\Omega,\mathcal{H},\hat{E})$ is called (centralized ) G-normal distributed if*

[TABLE]

where $\bar{X}$ is an independent copy of $X$ ($\bar{X}\stackrel{{\scriptstyle d}}{{=}}X$ and $\bar{X}$ is independent to $X$).

3 Convergence rates in the law of large numbers

In this section, we will study convergence rates in the law of large numbers under sublinear expectations. Let $(\Omega,\mathcal{H},\hat{E})$ be a sublinear expectation space as introduced in Section 2, and $\{X,X_{n},n\geq 1\}$ be a sequence of random variables in $\mathcal{H}$. We have the following convergences:

(1)

$\{X_{n},n\geq 1\}$ is said to quasi surely converge to $X$, if there exists a set $N\subset\Omega$ such that $\hat{E}[I_{N}]=0$ and $\forall\omega\in\Omega\backslash N$, $\lim_{n\to\infty}X_{n}(\omega)=X(\omega)$, which is denoted by $X_{n}\stackrel{{\scriptstyle q.s}}{{\longrightarrow}}X$.

(2)

$\{X_{n},n\geq 1\}$ is said to converge to $X$ in capacity, if for all $\varepsilon>0$, $\lim_{n\to\infty}V(\{|X_{n}-X|\geq\varepsilon\})=0$, which is denoted by $X_{n}\stackrel{{\scriptstyle V}}{{\longrightarrow}}X$.

(3)

$\{X_{n},n\geq 1\}$ is said to $L^{p}$ converge to $X~{}(p>0)$, if $\lim_{n\to\infty}\hat{E}[|X_{n}-X|^{p}]=0$, which is denoted by $X_{n}\stackrel{{\scriptstyle L^{p}}}{{\longrightarrow}}X$.

(4)

$\{X_{n},n\geq 1\}$ is said to completely converge to $X$, if for any $\varepsilon>0$, $\sum_{n=1}^{\infty}V(\{|X_{n}-X|\geq\varepsilon\})<\infty$, which is denoted by $X_{n}\stackrel{{\scriptstyle c.c.}}{{\longrightarrow}}X$.

(5)

$\{X_{n},n\geq 1\}$ is said to $S\mbox{-}L^{p}$ converge to $X~{}(p>0)$, if $\sum_{n=1}^{\infty}\hat{E}[|X_{n}-X|^{p}]<\infty$, which is denoted by $X_{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}X$.

(6)

$\{X_{n},n\geq 1\}$ is said to strongly quasi surely converge to $X$ with order $\alpha$ ($\alpha>0$), if $\sum_{n=1}^{\infty}|X_{n}-X|^{\alpha}<\infty$ q.s., which is denoted by $X_{n}\stackrel{{\scriptstyle S_{\alpha}\mbox{-}q.s.}}{{\longrightarrow}}X$.

Generally, we have

[TABLE]

If $\hat{E}$ has the monotone continuity property ([10, Definition 2.2(vii)]), i.e. for any $X_{n}\downarrow 0$ on $\Omega$, $\hat{E}[X_{n}]\downarrow 0$, then we have (see Hu and Zhou (2019))

[TABLE]

Let $\{X_{n},n\geq 1\}$ be a sequence of i.i.d. random variables such that $\hat{E}[X_{n}]=-\hat{E}[-X_{n}]=\mu$. Denote $S_{n}=X_{1}+X_{2}\cdots+X_{n}$, $\tilde{S}_{n}=\sum_{i=1}^{n}(X_{i}-\hat{E}[X_{i}])$. Then $\frac{S_{n}}{n}-\mu=\frac{\tilde{S}_{n}}{n}.$

3.1 Strong $L^{p}$ convergence version of the law of large numbers

Theorem 3.1

Suppose that $\hat{E}[|X_{1}|^{\alpha}]<\infty$ for some $\alpha>2$. We have

(i) if $0<p\leq 2$, then $\frac{S_{n}}{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\nrightarrow}}\mu;$

(ii) if $2<p\leq\alpha$, then $\frac{S_{n}}{n}\stackrel{{\scriptstyle S\mbox{-}L^{p}}}{{\longrightarrow}}\mu.$

To prove Theorem 3.1, we need one lemma.

Lemma 3.2

*(Theorem II.3.3, Lemma II.3.9 of Peng (2010), Theorem 3.2 of Hu (2011))

Let $\{X_{i}\}_{i=1}^{\infty}$ be a sequence of $\mathbb{R}^{d}$-valued i.i.d. random variables, satisfying $\hat{E}[X_{i}]=-\hat{E}[-X_{i}]=0$ and $\hat{E}[|X_{i}|^{2+\beta}]<\infty$ for some $\beta>0$. Then the sequence $\{\overline{S_{n}}\}$ defined by $\bar{S}_{n}:=(\sum_{i=1}^{n}X_{i})/\sqrt{n}$ converges in law to $\xi$, i.e.,*

[TABLE]

for any continuous function $\varphi\in C(\mathbb{R}^{d})$ satisfying the growth condition that $|\varphi(x)|\leq C(1+|x|^{p})$ for some constants $C>0,~{}p>0$, where $\xi$ is G-normal distributed with the law $N(\{0\}\times[\underline{\sigma}^{2},\bar{\sigma}^{2}])$, $\underline{\sigma}^{2}=-\hat{E}[-X_{i}^{2}],\bar{\sigma}^{2}=\hat{E}[X_{i}^{2}]$.

Proof of Theorem 3.1. Set $\tilde{X}_{i}=X_{i}-\mu$, $i\geq 1$. Then we have $\hat{E}[\tilde{X}_{i}]=\hat{E}[-\tilde{X}_{i}]=0,\forall i\geq 1$.

(i) Let $\hat{E}[\tilde{X}_{i}^{2}]=\bar{\sigma}^{2}$. Then $\bar{\sigma}^{2}>0$ by the assumption. By the positive homogeneity of $\hat{E}$, we have

[TABLE]

By Lemma 3.2, we have

[TABLE]

where $\xi$ is $G$-distributed with $\hat{E}[\xi^{2}]=\bar{\sigma}^{2}>0$. It follows that $V(\{\omega\in\Omega||\xi(\omega)|>0\})>0$, which implies that $\hat{E}[|\xi|^{p}]>0$ for any $p>0$. Denote $c_{p}=\hat{E}[|\xi|^{p}]$. Then by (3.2), there exists $N\in\mathbb{N}$ such that

[TABLE]

Therefore, for any $0<p\leq 2$, we have

[TABLE]

(ii) By the assumption, we know that for any $k\geq 1$, $X_{k+1}$ is independent to $(X_{1},\ldots,X_{k})$, which implies that $X_{k+1}$ is negatively dependent to $(X_{1},\ldots,X_{k})$ (see Zhang (2016, Definition 1.5) for the definition of negative dependence). By the Marcinkiewicz-Zygmund inequality under sublinear expectations (see Zhang (2016, (2.13)), the fact that $\hat{E}[\tilde{X}_{i}]=\hat{E}[-\tilde{X}_{i}]=0,\forall i\geq 1$, and Minkowski’s inequality (see Peng (2010, Proposition I.4.2), we have

[TABLE]

where $x^{+}=\max\{0,x\}$, $x^{-}=\max\{0,-x\}$, and $C_{\alpha}$ is a positive constant depending only on $\alpha$. It follows that

[TABLE]

For any $2<p<\alpha$, by Hölder’s inequality under sublinear expectations (see Peng (2010, Proposition I.4.2)) and (3.3), we obtain that

[TABLE]

In Chow (1988), the author also obtained the following result. Let $\{X,X_{n},n\geq 1\}$ be a sequence of i.i.d. random variables with $E[X]=0$ in a probability space $(\Omega,\mathcal{F},P)$. Suppose that $1<\alpha<2$. If $E[|X|^{\alpha}\log^{+}|X|]<\infty$, then

[TABLE]

As a consequence of Theorem 3.1 and its proof, we obtain the following two corollaries.

Corollary 3.3

Suppose that $\alpha>2$, $\hat{E}[|X_{1}|^{\alpha}]<\infty$, and $\hat{E}[X_{1}]=-\hat{E}[-X_{1}]=0$. Then, for any $2<p\leq\alpha$ and $\beta>(p+2)/2,$ we have

[TABLE]

Corollary 3.4

Suppose that $X_{1}\not\equiv\mu$ q.s., $\hat{E}[|X_{1}|^{\alpha}]<\infty$ for any $\alpha>0$, and $p>0$. Then

[TABLE]

3.2 Strongly quasi sure convergence version of the law of large numbers

Proposition 3.5

Suppose that $\hat{E}[|X_{1}|^{\alpha}]<\infty$ for some $\alpha>2$. Then for any $\beta>2$, we have

[TABLE]

i.e., $\frac{S_{n}}{n}\stackrel{{\scriptstyle S_{\beta}\mbox{-}q.s.}}{{\longrightarrow}}\mu$.

Proof. By Theorem 3.1(ii), we know that for $2<\beta\leq\alpha$, it holds that $\frac{S_{n}}{n}\stackrel{{\scriptstyle S\mbox{-}L^{\beta}}}{{\longrightarrow}}\mu,$ i.e.

[TABLE]

which together with the monotone convergence theorem (Cohen et al. (2011)) and the sublinear property implies that

[TABLE]

It follows that

[TABLE]

By the strong law of large numbers (see Theorem 1 of Chen (2016)), there exists a set $N\subset\Omega$ such that $V(N)=0$ and for any $\omega\in\Omega\backslash N$, there exists $M(\omega)\in\mathbb{N}$ such that for any $n\geq M(\omega)$,

[TABLE]

It follows that for $\beta>\alpha$ and $\omega\in\Omega\backslash N$,

[TABLE]

which together with (3.4) implies that for any $\beta>\alpha$,

[TABLE]

4 Some questions

In this section, we present some questions for further research.

Question 1. Can we extend the Hsu-Robbins theorem from a probability space to a sublinear expectation space?

Question 2. If the answer to Question 1 is affirmative, can we prove the converse result?

In fact, we can ask more questions. As to the results on the convergence rates in the law of large numbers in a probability space, we can consider the corresponding questions in a sublinear expectation space.

Acknowledgments  This work was supported by National Natural Science Foundation of China (Grant No. 11771309, 11871184).