Exact convergence rate in the Erdös-Rényi law of large numbers for independent non-identically distributed random variables (Q1901838)

Consider a sequence \(X_1,X_2,\dots\) of non-degenerate, centered, independent, but not necessarily identically distributed random variables. Extending the classical Erdös-Rényi law of large numbers [\textit{P. Erdös} and \textit{A. Rényi}, J. Anal. Math. 23, 103-111 (1970; Zbl 0225.60015)], the author obtains precise a.s. asymptotics for the maximum increments of the corresponding partial sum sequence \(\{S_n: n= 0, 1,\dots\}\). In fact the results improve on an earlier generalization of the Erdös-Rényi law to the non-i.i.d. case due to \textit{Z. Lin} [Chin. Ann. Math., Ser. B 11, No. 3, 376-383 (1990; Zbl 0715.60040)]. They actually extend the exact convergence rate statements of \textit{P. Deheuvels}, \textit{L. Devroye} and \textit{J. Lynch} [Ann. Probab. 14, 209-223 (1986; Zbl 0595.60033)] to this general case, assuming that the cumulant generating functions of the \(X_i\)'s are uniformly bounded in a circle around 0, and that, for some positive constants \(b_0\), \(b_1\), one has \(b_0\leq B_{n/k}/k\leq b_1\) for all \(n\) and \(k\) large, where \(B_{nk}= \text{Var}(S_{n+ k}- S_n)\). Proofs are omitted.