Marcinkiewicz–Zygmund Laws of Large Numbers under Sublinear Expectation

Cheng Hu

doi:10.1155/2020/5050973

Abstract

In this paper, we derive the weak and strong results of Marcinkiewicz–Zygmund laws of large numbers under sublinear expectation. The results are extensions of the Kolmogorov laws of large numbers under sublinear expectation and the classical Marcinkiewicz–Zygmund laws of large numbers.

Highlights

Introduction e theory of sublinear expectation was initiated by Peng [1, 2] to describe the probability uncertainties in statistics, economics, finance, and other fields which are difficult to be handled by the classical probability theory. e classical laws of large numbers (LLNs for short) which reveal the almost sure laws of stabilized partial sum are of great significance in the probability theory
Hu [8] improved the above results under a general moment condition for sublinear expectation which is the weakest one for sublinear expectation
E classical Marcinkiewicz–Zygmund strong LLNs generalized the Kolmogorov strong LLNs by extending the convergence rate of partial sum and give the relation between moment conditions and convergence rate. e norming constants become n1/p(0 < p ≤ 2) instead of n and the moment conditions depend on p

Summary

Preliminaries

Let (Ω, F) be a measurable space and H be a linear space of real functions defined on (Ω, F) such that if X1, X2, . . . , Xn ∈ H φ(X1, X2, . . . , Xn) ∈ H for each φ ∈ Cl,Lip(Rn) where Cl,Lip(Rn) denotes the linear space of local Lipschitz continuous functions φ satisfying. Xn) ∈ H for each φ ∈ Cl,Lip(Rn) where Cl,Lip(Rn) denotes the linear space of local Lipschitz continuous functions φ satisfying. We denote Cb,Lip(Rn) as the linear space of bounded Lipschitz continuous functions φ satisfying. A function E: H ⟶ R is said to be a sublinear expectation if it satisfies for ∀X, Y ∈ H,. For any random variable X ∈ F, the upper expectation defined by E􏽢[X]: supQ∈PEQ[X] is a sublinear expectation. Peng [2] gave the definition of identical distribution under the sublinear expectation space. E results in this paper do not need the random variables to be identically distributed. We consider the sequence 􏼈Xn􏼉∞ n 1 of independent random variables defined on a sublinear expectation space (Ω, H, E) with E[Xn] μ, E[Xn] μ for each n ≥ 1.

The Marcinkiewicz–Zygmund Weak LLNs

The Marcinkiewicz–Zygmund Strong LLNs