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

TL;DR

This paper establishes convergence rates in the law of logarithm for negatively dependent random variables under sub-linear expectations, extending classical results to a more general setting with dependent variables.

Contribution

It generalizes the law of logarithm convergence rates for negatively dependent variables under sub-linear expectations, a broader framework than classical probability.

Findings

Convergence of series involving sums of negatively dependent variables.

Extension of classical law of logarithm results to sub-linear expectation framework.

Conditions under which the convergence holds for zero-mean, finite variance variables.

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 $Lt=\ln \max\{\me,t\}$, $\psi(t)=\int_{1}^{t}(sh(s))^{-1}\dif s$, $t\ge 1$. In this sequel, we establish that $\sum_{n=1}^{\infty}(nh(n))^{-1}\vv\left\{|S_n|\ge (1+\varepsilon)\sigma\sqrt{2nL\psi(n)}\right\}<\infty$, $\forall \varepsilon>0$ if $\ee(X)=\ee(-X)=0$ and $\ee(X^2)=\sigma^2\in (0,\infty)$. The result generalizes that of NA random variables in probability space.