Theorems of complete convergence and complete integral convergence for END random variables under sub-linear expectations

Ziwei Liang,Qunying Wu

doi:10.1186/s13660-019-2064-0

Ziwei Liang, Qunying Wu

Open Access

https://doi.org/10.1186/s13660-019-2064-0

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Abstract

The goal of this paper is to build complete convergence and complete integral convergence for END sequences of random variables under sub-linear expectation space. By using the Markov inequality, we extend some complete convergence and complete integral convergence theorems for END sequences of random variables when we have a sub-linear expectation space, and we provide a way to learn this subject.

Highlights

Classical probability theorems were widely used in many fields, which only hold on some occasions of model certainty
Sublinear expectation and capacity are not additive, the limit theorems of classical probability space are no longer valid
Since the general framework of sub-linear expectation was introduced by Peng Shige, many scholars have paid close attention to it, and lots of excellent results have been established

Summary

Introduction

Classical probability theorems were widely used in many fields, which only hold on some occasions of model certainty. There are uncertainties, such as measures of risk, non-linear stochastic calculus and statistics in the process of finance At this time, sublinear expectation and capacity are not additive, the limit theorems of classical probability space are no longer valid. The sub-linear expectation axiom system makes up for the deficiency of limit theorems of classical probability space. Some corresponding results were obtained by Gut and Stadtmuller [18], Qiu and Chen [19], Wu and Jiang [20] and Feng and Wang [21], we still need to perfect the complete convergence and complete integral convergence under sub-linear expectation. We establish the complete convergence and complete integral convergence for END random variables under sub-linear expectation and generalize them [22] to the sub-linear expectation space. Non-increasing) functions, {fn(Xn); n 1} is a sequence of END random variables.

Because of hn kn

Noting that g

We know

3Bn dt