A note on records in a random sequence

Abstract

In an infinite sequence of independent identically distributed continuous random variables we study the number of strings of two subsequent records interrupted by a given number of non-records. By embedding in a marked Poisson process we prove that these counts are independent and Poisson distributed. Also the distribution of the number of uninterrupted strings of records is considered.

Non-Markovian random walks with memory lapses

“…However, as an illustration, we focus in two particular cases and compare their behaviours. The results are obtained by using techniques from recent works in the study of pattern strings in Bernoulli sequences (see, for instance [13,14,15,20]).…”

“…We consider the assumptions i.i.d. and the collection λ i = a/(a + b + i − 1) for a > 0, b ≥ 0 and i ≥ 1, usually denoted by Bern(a, b) (see [14,20]).…”

“…For instance, the sequence Bern(1, 0) arises in the limit in the study of cycles in random permutations and record values of continuous random variables. Moreover, the sequence Bern(a, 0) has some applications in nonparametric Bayesian inference and species allocation models (see [14,20] and references therein).…”

On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

“…Some relevant contributions on the subject are the works of Sarkar et al [12], Sen and Goyal [13], Holst [4], Dafnis et al [14], Huffer and Sethuraman [15], and Makri and Psillakis [16]. Applications of constrained ( , ℓ) strings of the general ( ≤ ≤ ℓ) or the restricted forms ( ≤ ℓ, = , ≥ ) were found in information theory and data compression (see Zehavi and Wolf [17], Jacquet and Szpankowski [18], and Stefanov and Szpankowski [19]) in urn models, record models, and random permutations (see Chern et al [20], Joffe et al [21], Chern and Hwang [22], Holst [5,9,10,23], and Huffer et al [24]) in system reliability (see Eryilmaz and Zuo [25], Eryilmaz and Yalcin [6], and Makri [26]) and in biomedical engineering (see Dafnis and Philippou [27]).…”

“…Special cases of PEUM are models of a (F/R − TM) fixed/random threshold (see, e.g., Eryilmaz and Yalcin [6], Makri and Psillakis [7], and Eryilmaz et al [8]), whereas a special case of HPUM is the (RIM) record indicator model (see, e.g., Holst [5,9,10], Demir and Eryılmaz [11], and Makri and Psillakis [7]). F/R − TM and RIM find potential applications in the frequency analysis and risk managing of the occurrence of critical events (records, extremes, and exceedances) in several scientific disciplines like physical sciences (e.g., seismology, meteorology, and hydrology) and stochastic financial analysis (e.g., insurance and financial engineering).…”