On the Law of Large Numbers for non-equally distributed weakly dependent random variables

TL;DR

This work extends the Weak Law of Large Numbers to sequences of weakly dependent and non-identically distributed random variables, possibly with infinite mean. It develops three LLN variants: Theorem 1 leverages Cesàro Uniform Integrability (RI by Cesàro) with conditional means $a_n$ to show $\mathsf E|S_n/n - a_n/n| \to 0$ and, when $a_n/n \stackrel{\mathsf P}{\to} 0$, $S_n/n \stackrel{\mathsf P}{\to} 0$; Theorem 2 imposes a weaker dependence condition via $\mathsf E(\xi_k|\xi_1+\dots+\xi_{k-1})$ and shows $S_n/n \stackrel{\mathsf P}{\to} 0$ under $\frac{1}{n}\sum_{k=1}^n \mathsf E|\mathsf E(\xi_k|\xi_1+\dots+\xi_{k-1})| \to 0$; Theorem 3 handles non-identically distributed cases without finite means by truncation and a Uniform Integrability condition on the auxiliary functions $\psi_n$, yielding $S_n/n \stackrel{\mathsf P}{\to} 0$. The results generalize classical Khintchine–Chow–Chandra LLNs under weak dependence and heterogeneous tails, using truncation, telescoping, and UI criteria to accommodate heavy-tailed or infinite-mean settings.

Abstract

Three versions of the Weak Law of Large Numbers are proposed for weakly dependent and generally speaking non-equally distributed random variables, with finite or possibly infinite expectations.