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.