Asymptotic behavior of the increments of sums of independent random variables (Q1901885)

In the paper reviewed above, the author provided a.s. convergence rates in a generalization of the Erdös-Rényi law of large numbers [\textit{P. Erdös} and \textit{A. Rényi}, J. Anal. Math. 23, 103-111 (1970; Zbl 0225.60015)] to the case of independent, but not necessarily identically distributed summands, when a kind of uniform Cramér-type condition is fulfilled. In this work, he states some analogous results for related maximum increments statistics which have been considered by other authors mainly in the i.i.d. case [cf. e.g. \textit{L. A. Shepp}, Ann. Math. Stat. 35, 419-423 (1964; Zbl 0146.39103) and ibid. 35, 424-428 (1964; Zbl 0146.39101), \textit{M. Csörgö} and the reviewer, Ann. Probab. 9, 988-996 (1981; Zbl 0477.60034), \textit{P. Deheuvels} and \textit{L. Devroye}, ibid. 15, 1363-1386 (1987; Zbl 0637.60039), \textit{Z. Lin}, Chin. Ann. Math., Ser. B 11, No. 3, 376-383 (1990; Zbl 0715.60040)]. Moreover, the author obtains a version of the Erdös-Rényi law when Cramér's condition is not satisfied. Proofs are omitted.