On the law of large numbers for nonidentically distributed weakly dependent summands

We propose new versions of the weak law of large numbers (LLN) for weakly dependent (nonidentically distributed, in general) summands, either assuming that each summand has finite expectation or without this assumption. One of the main conditions in the first of the three cases we consider (this case develops the ideas of Y. S. Chow [Ann. Math. Statist., 42 (1971), pp. 393–394]) is the Cesàro uniform integrability of the summands in the spirit of T. K. Chandra's studies of 1989–2012 on LLNs for pairwise independent random variables. In this result, the pairwise dependence conditions are replaced by quite different weak dependence type conditions in the spirit of A. N. Kolmogorov's paper [Atti Accad. Naz. Lincei Rend. (6), 9 (1929), pp. 470–474]; the only difference is that, in the present paper, we impose conditions only on the first moments of some conditional expectations, rather than on their second moments. In the second case, which is based on a slightly different weak dependence condition, we use the method of telescopic expansion and the fact that the convergence in probability to a constant can be interpreted as weak convergence. In the third case, we establish an LLN for summands without finite expectations, assuming, again, that they are not necessarily identically distributed.