On the integrability of \(\sup | S_ n| /n^{1/r}\) for \(1<r<2\)
type: Item
instance of: scholarly article
title: On the integrability of \(\sup | S_ n| /n^{1/r}\) for \(1<r<2\)
zbMATH Open document ID: 0622.60035
DOI: 10.1016/0167-7152(87)90103-9
author: Bong Dae Choi
author: Soo Hak Sung
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be i.i.d. random variables with \(EX_ 1=0\) and put \(S_ n=X_ 1+...+X_ n\). It is proved that for \(1<r<2\) the following three conditions are equivalent: \[ E(\sup_{n}n^{-1/r}| S_ n|)<\infty,\quad E(\sup_{n}n^{-1/r}| X_ n|)<\infty \quad and\quad E| X_ 1|^ r<\infty. \] It is stated that a straightforward modification of the proofs gives the equivalence, for \(1\leq r<2\), of the conditions \[ E(\sup_{n}n^{-1}| S_ n|^ r)<\infty,\quad E(\sup_{n}n^{-1}| X_ 1|^ r)<\infty \quad and\quad E\{| X_ 1|^ r(\log | X_ 1|)_+\}<\infty. \] Proofs are based on lemmas giving the equivalence of the condition for \(S_ n\) and the analogous condition for \(S_{n(k)}\) where \(n(k)<n(k+1)\) and \(\liminf_{k\to \infty}n(k-1)/n(k)>0\). \textit{A. Gut} [Ann. Probab. 7, 1059-1065 (1979; Zbl 0428.60036)] gives related results [see also \textit{A. Gut}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 46, 205-220 (1979; Zbl 0373.60059)].
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4009432
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: partial sums
zbMATH Keywords: suprema
zbMATH Keywords: moments
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90103-9
OpenAlex ID: W1988325136
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description: scientific article; zbMATH DE number 4009432