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 elect… Show more
“…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].…”
Section: δ-Recordssupporting
confidence: 94%
“…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.…”
Section: δ-Recordsmentioning
confidence: 99%
“…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.…”
Section: δ-Recordsmentioning
confidence: 99%
“…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].…”
Section: δ-Recordsmentioning
confidence: 99%
“…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].…”
Abstract. In recent years there has been a surge of interest in the statistics of recordbreaking events in stochastic processes. Along with that, many new and interesting applications of the theory of records were discovered and explored. The record statistics of uncorrelated random variables sampled from time-dependent distributions was studied extensively. The findings were applied in various areas to model and explain record-breaking events in observational data. Particularly interesting and fruitful was the study of record-breaking temperatures and their connection with global warming, but also records in sports, biology and some areas in physics were considered in the last years. Similarly, researchers have recently started to understand the record statistics of correlated processes such as random walks, which can be helpful to model record events in financial time series. This review is an attempt to summarize and evaluate the progress that was made in the field of record statistics throughout the last years.
“…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].…”
Section: δ-Recordssupporting
confidence: 94%
“…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.…”
Section: δ-Recordsmentioning
confidence: 99%
“…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.…”
Section: δ-Recordsmentioning
confidence: 99%
“…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].…”
Section: δ-Recordsmentioning
confidence: 99%
“…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].…”
Abstract. In recent years there has been a surge of interest in the statistics of recordbreaking events in stochastic processes. Along with that, many new and interesting applications of the theory of records were discovered and explored. The record statistics of uncorrelated random variables sampled from time-dependent distributions was studied extensively. The findings were applied in various areas to model and explain record-breaking events in observational data. Particularly interesting and fruitful was the study of record-breaking temperatures and their connection with global warming, but also records in sports, biology and some areas in physics were considered in the last years. Similarly, researchers have recently started to understand the record statistics of correlated processes such as random walks, which can be helpful to model record events in financial time series. This review is an attempt to summarize and evaluate the progress that was made in the field of record statistics throughout the last years.
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