On the rates of convergence for sums of dependent random variables

TL;DR

This paper establishes strong laws of large numbers for dependent nonnegative random variables, providing conditions under which normalized sums converge almost surely, including for pairwise negatively quadrant dependent sequences.

Contribution

It introduces new sufficient conditions for almost sure convergence of sums of dependent variables, extending classical results to dependent and negatively dependent cases.

Findings

Provides a.s. convergence criteria for dependent variables

Includes results for pairwise negatively quadrant dependent sequences

Uses moment inequalities to establish convergence

Abstract

For a sequence $\{X_{n}, \, n \geqslant 1 \}$ of nonnegative random variables where $\max[\min(X_{n} - s,t),0]$, $t > s \geqslant 0$, satisfy a moment inequality, sufficient conditions are given under which $\sum_{k=1}^n (X_k - \mathbb{E} \, X_k)/b_n \overset{\mathrm{a.s.}}{\longrightarrow} 0$. Our statement allows us to obtain a strong law of large numbers for sequences of pairwise negatively quadrant dependent random variables under sharp normalising constants.