Abstract
In this paper, we investigate the central limit theorems for sub-linear expectation for a sequence of independent random variables without assumption of identical distribution. We first give a bound on the distance between the normalized sum distribution and G-normal distribution which can be used to derive the central limit theorem for sub-linear expectation under the Lindeberg condition. Then we obtain the central limit theorem for capacity under the Lindeberg condition. We also get the central limit theorem for capacity for summability methods under the Lindeberg condition.
Highlights
1 Introduction Peng [15] put forward the theory of sub-linear expectation to describe the probability uncertainties in statistics and economics which are difficult to be handled by classical probability theory
The purpose of this paper is to investigate the CLTs for sub-linear expectation for a sequence of independent random variables without assumption of identical distribution
For sub-linear expectation under the Lindeberg condition directly, which coincides with the result in Zhang [23]
Summary
Peng [15] put forward the theory of sub-linear expectation to describe the probability uncertainties in statistics and economics which are difficult to be handled by classical probability theory. Peng [16] initiated the CLT for sub-linear expectation for a sequence of i.i.d. random variables with finite (2 + α)-moments for some α > 0. Li and Shi [13] got a CLT for sub-linear expectation without assumption of identical distribution. Li [14] proved a CLT for sub-linear expectation for a sequence of m-dependent random variables. Zhang [22] gained a CLT for sub-linear expectation under a moment condition weaker than (2 + α)-moments. Zhang [23] established a martingale CLT and functional CLT for sub-linear expectation under the Lindeberg condition. The purpose of this paper is to investigate the CLTs for sub-linear expectation for a sequence of independent random variables without assumption of identical distribution. We first give a bound on the distance between the normalized sum distribution E[φ( Sn )] and
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