Distribution of the number of consecutive records

We study the distribution of the number ξn,r of r consecutive records in a sequence of n independent and identically distributed random variables from a common continuous distribution, or equivalently, in a random permutation of n elements. We show that the asymptotic distribution of ξn,r is Poisson for r = 1, 2 and non-Poisson for r ≥ 3. Precise asymptotic results are derived for four probability distances of the associated approximations: Fortet-Mourier, total variation, Kolmogorov, and point metric. In particular, the distributions of ξn,r have the specific property that the last three distances are asymptotically of the same behaviors for r ≥ 2. We also provide interesting combinatorial bijections for ξn,2 and compute explicitly the limiting law for ξn,3 in terms of Kummer’s confluent hypergeometric functions. AMS 1991 subject classifications. Primary 60C05; secondary 60G70 05E15 62G30.