Strong laws of large numbers for general random variables in sublinear expectation spaces

Abstract

In this paper, we obtain the equivalent relations between Kolmogorov maximal inequality and Hájek-Rényi maximal inequality both in moment and capacity types in sublinear expectation spaces. Based on these, we establish several strong laws of large numbers for general random variables and obtain the growth rate of the partial sums. In a first application, a strong law of large numbers for negatively dependent random variables is obtained. In a second application, we consider the normalizing sequence {log n} n≥1 and get some special limit properties in sublinear expectation spaces.

Equivalent Conditions of Complete p th Moment Convergence for Weighted Sums of I. I. D. Random Variables under Sublinear Expectations

“…e interested reader could refer to Xu and Zhang [8,9], Chen [10], Gao and Xu [11], Fang et al [12], Hu et al [13], Hu and Yang [14], Huang and Wu [15], Kuczmaszewska [16], Ma and Wu [17], Wang and Wu [18], Wu and Jiang [19], Yu and Wu [20], Zhang [21], Zhong and Wu [22], and references therein for more limit theorems under sublinear expectations.…”

Convergence for sums of i.i.d. random variables under sublinear expectations

“…Zhang [12][13][14] obtained important inequalities including exponential and Rosenthal's inequalities and studied Donsker's invariance principle under sublinear expectations. Inspired by the works of Zhang [12][13][14][15], Huang and Wu [5] and Zhong and Wu [16] studied some limits theorems under sublinear expectation space. Recently, under sublinear expectations, Wu [10] proved precise asymptotics for complete integral convergence, and Xu and Cheng [11] established precise asymptotics in the law of iterated logarithm.…”

Equivalent conditions of complete $p$-th moment convergence for weighted sums of i. i. d. random variables under sublinear expectations

“…Xu and Cheng [7] studied precise asymptotics in the law of iterated logarithm under sublinear expectations. The interested reader could refer to Xu and Zhang [8,9], Chen [10], Gao and Xu [11], Fang et al [12], Hu et al [13], Hu and Yang [14], Huang and Wu [15], Kuczmaszewska [16], Ma and Wu [17], Wang and Wu [18], Wu and Jiang [19], Yu and Wu [20], Zhang [21], Zhong and Wu [22] and references therein for more limit theorems under sublinear expectations.…”