This note examines the almost sure behaviour of sums of arbitrary random variables without any moment conditions. Special attention is paid to sequences of independent non-identically distributed random variables.

This note examines the connection between general moment conditions and the applicability of the strong law of large numbers to a sequence of identically distributed random variables. Some generalizations of the Marcinkiewicz-Zygmund theorem are obtained.

A new generalization of the Borel-Cantelli Lemma is obtained. Several earlier generalizations are special cases.

A generalization of the Erdös-Rényi formulation of the Borel-Cantelli lemma is obtained.