1 Introduction and notations
The general framework of the sublinear expectation space was introduced by Peng [
13] and it is a natural extension of the classical linear expectation space with the linear property being replaced by the subadditivity and positive homogeneity. This simple generalization provides a very flexible framework to model uncertainty problems in statistics or finance. Peng [
14] introduced the independent and identically distributed random variables,
G-normal distribution and obtained the weak law of large numbers and the central limit theorem in sublinear expectation spaces. Since then, many authors began to investigate the limit properties, especially the law of large numbers in sublinear expectation spaces. For instance, Chen, Wu and Li [
5] gave a strong law of large numbers for non-negative product independent random variables; Chen [
2] obtained the Kolmogorov strong law of large numbers for independent and identically distributed random variables; Zhang [
21] gave the necessary and sufficient conditions of Kolmogorov strong law of larger numbers holding for independent and identically distributed random variables under a continuous sublinear expectation; Zhang [
22] obtained a strong law of large numbers for a sequence of extended independent random variables; Chen, Hu and Zong [
3] gave several strong laws of large numbers under some moment conditions with respect to the partial sum and some conditions similar to Petrov’s; Chen, Huang and Wu [
4] established an extension of the strong law of large numbers under exponential independence. There are also many results on law of large numbers under different non-additive probability framework. The reader can see [
1,
6,
12,
18] and the references therein.
However, none of these papers investigate the growth rate of the partial sums. The main purpose of this paper is to explore the growth rate of the partial sums and investigate the strong law of large numbers for general random variables in sublinear expectation spaces.
In classical probability or expectation space, Fazekas and Klesov [
9] gave a general approach of obtaining the strong law of large numbers for sequences of random variables by using a Hájek–Rényi type maximal inequality. This general approach made no restriction on the dependence structure of random variables. Under the same conditions as those in Fazekas and Klesov [
9], Hu and Hu [
10] obtained the growth rate for the sums of random variables. Sung, Hu and Volodin [
17] improved these results and applied this approach to the weak law of large numbers for tail series. Tómács and Líbor [
20] used the probability Hájek–Rényi type maximal inequality to prove the strong law of large numbers. More results and applications of this approach in classical probability space can be found in Fazekas [
8] and the references therein.
In this paper, we will extend the approach used by Fazekas and Klesov [
9] and Hu and Hu [
10] to sublinear expectation spaces. In Sect.
2, we firstly show the equivalent relations between Kolmogorov maximal inequality and Hájek–Rényi maximal inequality both in the moment and capacity types in sublinear expectation spaces. Based on these, we establish several strong laws of large numbers for general random variables in sublinear expectation spaces and obtain the growth rate of the partial sums in Sect.
3. As an application, a strong law of large numbers for negatively dependent random variables is obtained since the Kolmogorov type maximal inequality for negatively dependent random variables has been proved in Sect.
2. In Sect.
4, we give another application. We consider the normalizing sequence
\(\{\log n\}_{n\ge 1}\) and get some limit properties in sublinear expectation spaces.
In the remainder of this section, we give some notations in sublinear expectation spaces. We use the framework and notations in Peng [
14] and [
15]. Let
\((\varOmega ,\mathcal{F})\) be a given measurable space and
\(\mathscr{H}\) be a linear space of real measurable functions defined on
\((\varOmega , \mathcal{F})\) such that if
\(X_{1},\ldots , X_{n} \in \mathscr{H}\) then
\(\psi (X_{1},\ldots ,X_{n})\in \mathscr{H}\), for each
\(\psi \in C_{l,\mathrm{Lip}}( \mathbb{R}^{n})\), where
\(C_{l,\mathrm{Lip}}(\mathbb{R}^{n})\) denotes the linear space of functions
ψ satisfying
$$\begin{aligned}& \bigl\vert \psi (y)-\psi (z) \bigr\vert \leq C\bigl(1+ \vert y \vert ^{m}+ \vert z \vert ^{m}\bigr) \vert y-z \vert , \\& \quad \mbox{for all } y,z\in \mathbb{R}^{n}, \mbox{for some } C>0, m \in \mathbb{N} \mbox{ depending on } \psi . \end{aligned}$$
Here and in the sequel,
\(\mathbb{N}\) denotes the set of all non-negative integers and
\(\mathbb{N}^{*}\) denotes the set of all positive integers. In general,
\(C_{l,\mathrm{Lip}}(\mathbb{R}^{n})\) can be replaced by
\(C_{b,\mathrm{Lip}}( \mathbb{R}^{n})\), which is the space of all bounded and Lipschitz continuous functions on
\(\mathbb{R}^{n}\). The space
\(\mathscr{H}\) is considered as a space of random variables.
Obviously, for all
\(X\in \mathscr{H}\),
\(\mathcal{E}[X]\le \mathbb{E}[X]\). Furthermore,
\((\mathbb{E}, \mathcal{E})\) can generate a pair
\((V,v)\) of capacities by
$$ V(A):=\sup_{\theta \in \varTheta } E_{\theta }[I_{A}],\qquad v(A):=1-V\bigl(A ^{c}\bigr), \quad \mbox{for all } A\in \mathcal{F}, $$
where
\(\{E_{\theta }: \theta \in \varTheta \}\) is the family of linear expectations in Lemma
1.2 and
\(A^{c}\) is the complement set of
A. It is easy to check that
V is subadditive, monotonic and
$$ \mathbb{E}[X]\le V(A)\le \mathbb{E}[Y], \qquad \mathcal{E}[X]\le v(A) \le \mathcal{E}[Y], \quad \mbox{if } X\le I_{A}\le Y, X,Y\in \mathscr{H}. $$
(1)
Moreover, if
\(I_{A}\in \mathscr{H}\), then
\(V(A)=\mathbb{E}[I_{A}]\) and
\(v(A)=\mathcal{E}[I_{A}]\).
2 Kolmogorov type and Hájek–Rényi type maximal inequalities
For convenience, let \(\max_{i\in A} a_{i}=0\) and \(\sum_{i\in A} a_{i}=0\) if \(A=\emptyset \).
Let \(\{X_{i}\}_{i\ge 1}\) denote a sequence of random variables in the sublinear expectation space \((\varOmega ,\mathscr{H},\mathbb{E})\) and the partial sums of random variables be \(S_{n}=\sum_{i=1}^{n}X_{i}\) for all \(n\in \mathbb{N}^{*}\) and \(S_{0}=0\). The constant c may be different in different places.
At the end of this section, we give two Kolmogorov type maximal inequalities for negatively dependent random variables in sublinear expectation spaces. Before we do this, we prove two propositions which will be used in the proof of Kolmogorov type maximal inequality.
The following proposition gives the Chebyshev inequalities in sublinear expectation spaces which have been proved in Chen, Wu and Li [
5]. But the proof there is not valid for the capacities defined by Zhang [
21] (see Remark
1.3 in this paper). We give a new proof here which is also valid for the capacities defined by Zhang [
21].
3 Strong laws of large numbers and the growth rate of partial sums
In this section and the sequel, we consider the sublinear expectation
\(\mathbb{E}\) can be represented by
$$ \mathbb{E}[X]=\sup_{P\in \mathcal{P}}E_{P}[X], \quad \mbox{for all } X \in \mathscr{H}, $$
where
\(\mathcal{P}\) is a nonempty set of
σ-additive probabilities on
\(\mathcal{F}\). It is easy to check that
$$ \mathcal{E}[X]=\inf_{P\in \mathcal{P}}E_{P}[X], \quad \mbox{for all } X \in \mathscr{H}, $$
and
\((V,v)\) can be rewritten as
$$ V(A)=\sup_{P\in \mathcal{P}}P(A),\qquad v(A)=\inf_{P\in \mathcal{P}}P(A),\quad \mbox{for all } A\in \mathcal{F}. $$
Clearly,
V is inner continuous and
v is outer continuous, that is,
\(V (A_{n})\uparrow V (A)\), if
\(A_{n}\uparrow A\); and
\(v(A_{n})\downarrow v(A)\), if
\(A_{n}\downarrow A\), where
\(A_{n}, A\in {\mathcal{F}}\),
\(n\ge 1\) (see Lemma 2.1 in Chen, Wu and Li [
5]). Theorem
3.2 shows that these properties also hold for
\(\mathbb{E}\) and
\(\mathcal{E}\).
Let
φ be a positive function satisfying
$$ \sum_{n=1}^{\infty } \frac{\varphi (n)}{n^{2}}< \infty \quad \mbox{and}\quad 0< \varphi (x)\uparrow \infty \quad \mbox{on } [c,\infty ) \mbox{ for some } c>0. $$
(8)
For convenience, we define
\(\frac{1}{0}=\infty \),
\(\frac{1}{\infty }=0\) and
\(\varphi (\infty )=\infty \).
Theorems
3.6,
3.8 and
3.9 give several strong laws of large numbers and the growth rate of the partial sums for general random variables in sublinear expectation spaces.
The result of Theorem
3.8 is more precise than Theorem
3.6. When the sublinear expectation space degenerates to the classical probability space, Theorem
3.8 and Theorem
3.9 give more precise results than Theorem 2.1 in Fazekas and Klesov [
9], Lemma 1.2 in Hu and Hu [
10], Theorem 3.4 in Tómács [
19] and Theorem 2.4 in Tómács and Líbor [
20].
4 An application to the logarithmically weighted sums
By using Theorem
3.6 and
3.8 to the logarithmically weighted sums, we can get Theorem
4.2, which sharpens the result of Theorem 8.1 in Fazekas and Klesov [
9] and Theorem 2.5 in Hu and Hu [
10] under the same condition in the classical probability theory and extends it to the sublinear expectation space. Some of our idea for obtaining Theorem
4.2 come from these papers.
Throughout the sequel of this section, the sublinear expectation spaces and the random variable sequence \(\{X_{n}\}_{n\ge 1}\) are further supposed to satisfy the following two assumptions.
A random variable
ξ is
G-normal distributed (denoted by
\(\xi \sim N(0, [\underline{ \sigma }^{2}, \overline{\sigma }^{2}])\)) under a sublinear expectation
\(\widetilde{\mathbb{E}}\), if and only if for any
\(f \in C_{b,\mathrm{Lip}}( \mathbb{R})\), the function
\(u(t,x)=\widetilde{\mathbb{E}} [f (x+\sqrt{t} \xi ) ]\) (
\(x\in \mathbb{R}\),
\(t\ge 0\)) is the unique viscosity solution of the following
G-heat equation:
$$ \textstyle\begin{cases} \partial _{t} u -G ( \partial _{xx}^{2} u ) =0, \quad (x,t)\in \mathbb{R}\times (0,\infty ), \\ u(0,x)=f(x), \end{cases} $$
where
\(G(a)=\frac{1}{2}\widetilde{\mathbb{E}}[a\xi ^{2}]\),
\(a\in \mathbb{R}\), is determined by the variances
\(\bar{\sigma }^{2}:= \widetilde{\mathbb{E}}[\xi ^{2}]\) and
\(\underline{\sigma }^{2}:= \widetilde{\mathcal{E}}[\xi ^{2}]\). If
\(\bar{\sigma }^{2}=\underline{ \sigma }^{2}\), then
G-normal distribution is just the classical normal distribution
\(N(0, \bar{\sigma }^{2})\).
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