Asymptotic behaviour of a number of repeated records

E. Khmaladze, M. Nadareishvili and A. Nikabadze

Statistics & Probability Letters, 1997, vol. 35, issue 1, 49-58

Abstract: Given a sequence of i.i.d. random variables with continuous distribution function we study the number of [var epsilon]-repetitions of the current record value from the time it occurs up to some n, as n --> [infinity]. This number of repetitions typically does not converge, but oscillates indefinitely. However, if it does converge, it can do so only to 1 or to [infinity]. For deterministic choices of [var epsilon], we give exact and asymptotic distributions and set conditions on the tail of F for convergence in probability to 1 and to [infinity] and for a.s. convergence to 1. For one choice of [var epsilon] as a radom variable, we obtain a never converging sequence of [var epsilon]-repetitions with limiting distribution being the so called Zipf's law, famous in an entirely different context.

Keywords: Regularly; varying; functions; Repetitions; of; record; Zipf's; law; Point; processes; Independent; random; walks (search for similar items in EconPapers)
Date: 1997
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Citations: View citations in EconPapers (9)

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