Distribution of the number of consecutive records. (English) Zbl 0969.60017

We study the distribution of the number \(\xi_{n,r}\) of times that \(r\) consecutive records occur 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 \(\xi_{n,r}\) exists and is Poisson for \(r=1,2\) and non-Poisson for \(r\geq 3\). 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 \(\xi_{n,r}\) have the specific property that the last three distances are asymptotically of the same behavior for \(r\geq 2\). We also provide interesting combinatorial bijections for \(\xi_{n,2}\) and compute explicitly the limiting law for \(\xi_{n,3}\) in terms of Kummer’s confluent hypergeometric functions.


60C05 Combinatorial probability
62E17 Approximations to statistical distributions (nonasymptotic)
60G70 Extreme value theory; extremal stochastic processes
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