Gaps in discrete random samples

Abstract

Let (X_i)_i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ₀ of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X₁,...,X_n} of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (q_k)_k∈ℕ0, q_k=P(X₁≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑_k=0^∞ q_k+1/q_k <∞ and lim_k→∞q_k+1/q_k=0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if q_k+1/q_k→ 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme--Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.