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July, 1987 The Contribution to the Sum of the Summand of Maximum Modulus
William E. Pruitt
Ann. Probab. 15(3): 885-896 (July, 1987). DOI: 10.1214/aop/1176992071

Abstract

Let $X_k$ be i.i.d., $S_n = X_1 + \cdots + X_n$, and $X^{(1)}_n$ the term of maximum modulus among $\{X_1,\ldots, X_n\}$. Let $u_k = P\{2^k < |X_1| \leq 2^{k+1}\parallel |X_1| > 2^k\}$. The main result is that $X^{(1)}_n/S_n \rightarrow 1$ a.s. $\operatorname{iff} \sum u^2_k < \infty$. Furthermore, for any positive integer $r, \lim\inf_{n\rightarrow\infty} |X^{(1)}_n/S_n| = r^{-1} \mathrm{a.s.} \operatorname{iff} \sum_k u^r_k = \infty$ and $\sum_ku^{r+1}_k < \infty$. If $\sum_ku^r_k = \infty$ for all $r$ then $\lim\inf_{n\rightarrow\infty} |X^{(1)}_n/S_n| = 0$ a.s.

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William E. Pruitt. "The Contribution to the Sum of the Summand of Maximum Modulus." Ann. Probab. 15 (3) 885 - 896, July, 1987. https://doi.org/10.1214/aop/1176992071

Information

Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0625.60031
MathSciNet: MR893904
Digital Object Identifier: 10.1214/aop/1176992071

Subjects:
Primary: 60F15
Secondary: 60G50

Keywords: dominance of maximal summand , Random walk , slowly varying tails , trimmed sums

Rights: Copyright © 1987 Institute of Mathematical Statistics

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Vol.15 • No. 3 • July, 1987
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