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April, 1974 On Convergence in $r$-Mean of Some First Passage Times and Randomly Indexed Partial Sums
Allan Gut
Ann. Probab. 2(2): 321-323 (April, 1974). DOI: 10.1214/aop/1176996712

Abstract

Let $S_n, n = 1,2, \cdots$ denote the partial sums of i.i.d. random variables with positive, finite mean and with a finite moment of order $r, 1 \leqq r < 2$. Let $Z_n, n = 1,2, \cdots$ denote the partial sums of i.i.d. random variables with a finite moment of order $r, 0 < r < 2$, and with mean 0 if $1 \leqq r < 2$. Let $N(c) = \min \{n; S_n > c\}, c \geqq 0$. Theorem 1 states that $N(c)$, (suitably normalized), tends to 0 in $r$-mean as $c \rightarrow \infty$. The first part of that proof follows by applying Theorem 2, which generalizes the known result $E|Z_n|^r = o(n)$, as $n\rightarrow \infty$ to randomly indexed partial sums.

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Allan Gut. "On Convergence in $r$-Mean of Some First Passage Times and Randomly Indexed Partial Sums." Ann. Probab. 2 (2) 321 - 323, April, 1974. https://doi.org/10.1214/aop/1176996712

Information

Published: April, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0278.60032
MathSciNet: MR358972
Digital Object Identifier: 10.1214/aop/1176996712

Subjects:
Primary: 60G40
Secondary: 60G45, 60G50, 60K05

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 2 • April, 1974
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