On Two Problems Involving Partial Sums

Abstract

Two problems are considered: (1) The expected waiting time for the partial sums of i.i.d. positive integer-valued random variables to be a multiple of k for the first time. For aperiodic distributions this expected value is shown to equal k. The periodic case is considered, as well as the waiting time for the partial sums to equal r modulo k. (2) For two independent sequences of partial sums of positive i.i.d. random variables, the expected value of the smallest common partial sum is derived. If both distributions are aperiodic, this expected value is the product of the two means. The periodic case is considered, as well as the case of more than two sequences.