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

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 X1, X2, ... be independent random variables with common distribution F. Let M(r) nbe the rth largest among X1, ..., Xn; also let X(r) nbe the observation with the rth largest absolute value among X1, ..., Xn. Set Sn= ∑n 1X i,(r)Sn= Sn- M(1) n- ⋯ - M(r) nand(r)S̃n= Sn- X(1) n- ⋯ - X(r) n((0)S̃n=(0)S̃n= Sn). We find simple criteria in terms of F for(r)Sn/Bn→ p ± ∞ (i.e.,(r)Sn/Bntends to ∞ or to -∞ in probability) or(r)S̃n/Bn→ p ± ∞ when r = 0, 1, ... 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)Sn/n → p ∞. Some equivalences for the divergence of |(r)S̃n|/|X(r) n|, or of(r)Sn/(X-) (s) n, where (X-) (s) nis the sth largest of the negative parts of the Xi, and for the convergence$P\{S_n > 0\}\rightarrow 1$, as n→∞, 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.