On complete convergence in the law of large numbers for subsequences
Let \(X_ 1,X_ 2,..\). be i.i.d., \(S_ n=X_ 1+...+X_ n\) and \(\{n_ k\}^ a \)strictly increasing subsequence of integers. Necessary and sufficient conditions for (*) \(\sum P(| S_{n_ k}/n_ k| >\epsilon)<\infty\) were given by the reviewer ad \textit{T. G. Kurtz} [Ann. Probab. 8, 176-182 (1980; Zbl 0426.60026)], motivated in part from some branching process applications. Their conditions were in terms of a test function \(\psi\) growing in some sense like \(\{n_ k\}\) and the proof relied on technical ideas from \textit{T. G. Kurtz} [Ann. Math. Stat. 43, 1874-1883 (1972; Zbl 0251.60019)]. In the present note, alternative criteria somewhat more directly related to the subsequence are given, as well as proofs which are somewhat more elementary [a main step is an application of an inequality due to \textit{J. Hoffmann-Jørgensen}, Stud. Math. 52, 159-186 (1974; Zbl 0265.60005)]. First those subsequences are determined for which mean zero is necessary and sufficient for (*), and then integrability conditions in terms of the growth of the subsequence are given in the case when a moment of order greater than one exists.
scientific article; zbMATH DE number 3932128