## The Annals of Probability

### Infinite Limits and Infinite Limit Points of Random Walks and Trimmed Sums

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 22, Number 3 (1994), 1473-1513.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988609

Digital Object Identifier
doi:10.1214/aop/1176988609

Mathematical Reviews number (MathSciNet)
MR1303651

Zentralblatt MATH identifier
0816.60067

JSTOR