The convergence of the sums of independent random variables under the sub-linear expectations

Abstract

Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables on a probability space $(\Omega ,\mathcal{F},P)$ and $S_n={\sum }_{k=1}^n{X}_k$. It is well-known that the almost sure convergence, the convergence in probability and the convergence in distribution of $S_n$ are equivalent. In this paper, we prove similar results for the independent random variables under the sub-linear expectations, and give a group of sufficient and necessary conditions for these convergence. For proving the results, the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established. As an application of the maximal inequalities, the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.

References in corpus (3)

• A New Central Limit Theorem under Sublinear Expectations
• Nonlinear Expectations and Stochastic Calculus under Uncertainty
• Tightness, weak compactness of nonlinear expectations and application to CLT

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• On the laws of the iterated logarithm under the sub-linear expectations without the assumption on the continuity of capacities
• On the law of the iterated logarithm under the sub-linear expectations
• Convergence rates in the law of large numbers under sublinear expectations
• Functional central limit theorems for random vectors under sub-linear expectations
• Pseudo-independence, independence and related limit theorems under sublinear expectations