Separation of the maxima in samples of geometric random variables

We consider samples of n geometric random variables ω ₁ ω ₂ · · · ω _n where ({ω _j = i}=pq ^i-1, 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.