Strong law of large numbers for weighted sums of m-widely acceptable random variables under sub-linear expectation space

Abstract

In this article, using the Fuk-Nagaev type inequality, we studied general strong law of large numbers for weighted sums of $ m $-widely acceptable ($ m $-WA, for short) random variables under sublinear expectation space with the integral condition $ \hat{\mathbb{E}} \left ( f^-\left ( \left | X \right | \right ) \right ) \le \mathrm{C}_\mathbb{V}\left ( f^-\left ( \left | X \right | \right ) \right )< \infty $ and $ Choquet $ integrals existence, respectively, where$ f\left ( x \right ) = x^{1/\beta }L\left ( x \right ) $for $ \beta > 1 $, $ L\left (x \right) > 0 $ $ \left(x > 0\right) $ was a monotonic nondecreasing slowly varying function, and $ f^-\left (x \right) $ was the inverse function of $ f\left(x\right) $. One of the results included the Kolmogorov-type strong law of large numbers and the partial Marcinkiewicz-type strong law of large numbers for $ m $-WA random variables under sublinear expectation space. Besides, we obtained almost surely convergence for weighted sums of $ m $-WA random variables under sublinear expectation space. These results improved the corresponding results of Ma and Wu under sublinear expectation space.

Keywords

• $ m $-widely acceptable random variables
• strong law of large numbers
• sublinear expectation space
• slowly varying function
• almost surely convergence