On the Strong Law of Large Numbers for Nonidentically Distributed, Partially Dependent Random Variables

A new version of the strong law of large numbers is proposed for a “good” sequence of pairwise independent random variables (r.v.’s) with a small portion of “bad” dependent r.v.’s. The main goal was to weaken the requirement for the existence of the expectation for each term: the r.v.’s of the bad, “sparse” part of the sequence may have moments of orders converging to zero.