On Consecutive Records in Certain Bernoulli Sequences

Abstract

In an infinite sequence of independent Bernoulli trials with success probabilities pk=a/(a+b +k-1) for k=1,2,3,…, let Nr be the number of r≥2 consecutive successes. Expressions for the first two moments of Nr are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is indicated how the distribution of Nr can be calculated for small r.

A note on records in a random sequence

“…As M 1 is Poisson with mean 1 we get for n>1 the unconditional probability Further results on consecutive records can be found in [3] and [10].…”

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]).…”

“…Formally, for n > l, and where second and third terms represent a memory lapse at the beginning and at the end of the sequence, respectively. In other words, M l (n) is the number of runs of 0's (see [13,15]) of length l in the first n trials of {Y i }.…”

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

“…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).…”