Separation of the maxima in samples of geometric random variables

We consider samples of $n$ geometric random variables $omega_1, omega_2 cdotsomega_n$ where $P{omega_j=i}=pq^{i-1}$, for $1leq j leq n$, 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.