Separation of the maxima in samples of geometric random variables

Abstract

We consider samples of n geometric random variables W1 W2 ... Wn where P{W) = i} = pqi-l, for 1 ? j ? n, with p + q = 1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice's method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random variables with at least two maxima.