Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations

Abstract

Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n={\sum }_{i=1}^n{X}_i$, $n\ge 1$. Assume that $h(\cdot )$ is a positive non-decreasing function on $(0,\infty )$ fulfulling $\int_{1}^{\infty}(th(t))^{-1}\dif t=\infty$. Write $Lt=\ln \max\{\me,t\}$, $ψ(t)=\int_{1}^{t}(sh(s))^{-1}\dif s$, $t\ge 1$. In this sequel, we establish that $\sum_{n=1}^{\infty}(nh(n))^{-1}\vv\left\{|S_n|\ge (1+\varepsilon)σ\sqrt{2nLψ(n)}\right\}<\infty$, $\forall \epsilon >0$ if $\ee(X)=\ee(-X)=0$ and $\ee(X^2)=σ^2\in (0,\infty)$. The result generalizes that of NA random variables in probability space.

8 pages, submitted to Mathematica Applicata