Weak laws for dependent sums

Abstract:

A general weak law of large numbers for sums ${S_n} = {X_1} + \cdots + {X_n}$ is proved. That is, without assuming the existence of any moments, and allowing any sort of dependence structure, conditions are given for ${S_n}/n \to 0$ in probability; the conditions are not necessary. However they are sufficient for a much stronger statement, namely that ${S_{{\nu _n}}}/{\nu _n} \to 0$ in probability in many cases where positive, integer-valued random variables ${\nu _n} \to \infty$.