Asymptotic normality for the counting process of weak records and δ-records in discrete models

Abstract

Let {Xn, n ≥ 1} be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call Xn a δ-record if Xn > max{X1, . . . , Xn−1} + δ, where δ is an integer constant. We use martingale arguments to show that the counting process of δ-records among the first n observations, suitably centered and scaled, is asymptotically normally distributed for δ = 0. In particular, taking δ = −1 we obtain a central limit theorem for the number of weak records. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 3, 754-781. This reprint differs from the original in pagination and typographic detail.

Records in stochastic processes—theory and applications

“…Instead of describing their rather complex derivations, we will now demonstrate an elementary approach that illustrates the asymptotic behavior of δ-records in time series of RV's from the three classes of EVS in the regime of small δ ≪ 1. Our findings are in good agreement with the results of Gouet et al [39].…”

“…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.…”

“…In particular Gouet et al made important progress on this problem. In [39], they discussed the process of δ-records in detail and, using a so-called Martingale approach (see [39] and references therein), proved a limit theorem for the asymptotic distribution of the record number in this case. Instead of describing their rather complex derivations, we will now demonstrate an elementary approach that illustrates the asymptotic behavior of δ-records in time series of RV's from the three classes of EVS in the regime of small δ ≪ 1.…”

“…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].…”