Let Sn,n = 1, 2, ..., denote the partial sums of integrable random variables. No assumptions about independence are made. Conditions for the finiteness of the moments of the first passage times N(c) = min {n: Snca(n)}, where c = 0 and a(y) is a positive continuous function on [0, [infinity]),...

Let X1, X2,... be independent random variables with a common continuous distribution function. Rates of convergence in limit theorems for record times and the associated counting process are established. The proofs are based on inversion, a representation due to Williams and random walk methods.

We extend the classical Hsu-Robbins-Erdos theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log+X)[alpha]}<[infinity] for some [alpha]>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that...</[infinity]>

We prove some analogs of results from renewal theory for random walks in the case when there is a drift, more precisely when the mean of the kth summand equals k[gamma][mu], k>=1, for some [mu]>0 and 0<[gamma]<=1.

Let X1, X2,..., be i.i.d. random variables, whose distribution function is continuous and set Yn = max{X1, X2,..., Xn}, n [greater-or-equal, slanted] 1. We provide elementary proofs of some results on the set of limit points of Yn.

The waiting times in certain multinomial experiments are studied. Special cases are the waiting times in a roulette game and in some dice problems sought for earlier. The main ingredients are the theory of stopped sums and an alternative formulation using independent Poisson processes.