Central limit theorems for the number of records in discrete models

Abstract

Consider a sequence (X n ) of independent and identically distributed random variables taking nonnegative integer values, and call X n a record if X n > max{X 1 , . . . , X n−1 }. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.

Records in stochastic processes—theory and applications

“…The concept of δ-records (or near-records) was discussed by various authors, for instance by Gouet et al [38,39,58] or Balakrishnan et al [59,60]. In particular Gouet et al made important progress on this problem.…”

“…As in the case of rounding discussed before, the δ is negligible in the Fréchet class and has a strong effect that increases with n in the Weibull class. It is straightforward to show that, in the Weibull class, the record rate will eventually decay exponentially, which leads to a finite asymptotic record number [38,39].…”

“…Even though most of the classical theory is developed for random numbers sampled from continuous distributions, practical measurements are always imprecise and rounded to a certain accuracy. Both the record statistics of random numbers from discrete distributions [36,37,38,39] as well as the consequences of analyzing records in time series of random numbers that were drawn from continuous distributions and then discretizes in a measuring process were discussed in recent years [40].…”

Asymptotic normality for the number of records from general distributions

“…In Section 4, distributions for which N n is not asymptotically normal are characterized in terms of their discrete and continuous components. Observe that in Corollary 1 no conditions are imposed on the hazard rates r n so this result extends Theorem 1 of [6]. Also, Corollary 1 gives a positive answer to a question raised in [2, Section 1, p. 323].…”

Laws of large numbers for the number of weak records

“…On the other hand, it is known that r k → 1 implies m(n log n) − m(n) − 1 < γ log log n, for some γ > 0 and all large enough n, (see page 789 of Gouet et al (2005)). Thus, by (13),…”