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July, 1994 Infinite Limits and Infinite Limit Points of Random Walks and Trimmed Sums
Harry Kesten, R. A. Maller
Ann. Probab. 22(3): 1473-1513 (July, 1994). DOI: 10.1214/aop/1176988609


We consider infinite limit points (in probability) for sums and lightly trimmed sums of i.i.d. random variables normalized by a nonstochastic sequence. More specifically, let $X_1, X_2, \ldots$ be independent random variables with common distribution $F$. Let $M^{(r)}_n$ be the $r$th largest among $X_1, \ldots, X_n$; also let $X^{(r)}_n$ be the observation with the $r$th largest absolute value among $X_1, \ldots, X_n$. Set $S_n = \sum^n_1X_i, ^{(r)}S_n = S_n - M^{(1)}_n - \cdots - M^{(r)}_n$ and $^{(r)}\tilde{S}_n = S_n - X^{(1)}_n - \cdots - X^{(r)}_n (^{(0)}\tilde{S}_n = ^{(0)}\tilde{S}_n = S_n)$. We find simple criteria in terms of $F$ for $^{(r)}S_n/B_n \rightarrow p \pm \infty$ (i.e., $^{(r)}S_n/B_n$ tends to $\infty$ or to $-\infty$ in probability) or $^{(r)}\tilde{S}_n/B_n \rightarrow p \pm \infty$ when $r = 0, 1, \ldots$. Here $B_n \uparrow \infty$ may be given in advance, or its existence may be investigated. In particular, we find a necessary and sufficient condition for $^{(r)}S_n/n \rightarrow p \infty$. Some equivalences for the divergence of $|^{(r)}\tilde{S}_n|/|X^{(r)}_n|$, or of $^{(r)}S_n/(X^-)^{(s)}_n$, where $(X^-)^{(s)}_n$ is the $s$th largest of the negative parts of the $X_i$, and for the convergence $P\{S_n > 0\}\rightarrow 1$, as $n\rightarrow\infty$, are also proven. In some cases we treat divergence along a subsequence as well, and one such result provides an equivalence for a generalized iterated logarithm law due to Pruitt.


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Harry Kesten. R. A. Maller. "Infinite Limits and Infinite Limit Points of Random Walks and Trimmed Sums." Ann. Probab. 22 (3) 1473 - 1513, July, 1994.


Published: July, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0816.60067
MathSciNet: MR1303651
Digital Object Identifier: 10.1214/aop/1176988609

Primary: 60F15
Secondary: 60F05 , 60J15 , 62G30

Keywords: infinite limit points , order statistics , Relative stability , trimmed sums

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 3 • July, 1994
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