Near-Record Values in Discrete Random Sequences

Lafuente, Miguel ; Gouet, Raúl ; López, F. Javier ; Sanz, Gerardo

Resumen: Given a sequence (Xn) of random variables, Xn is said to be a near-record if Xn∈(Mn−1−a,Mn−1], where Mn=max{X1,…,Xn} and a>0 is a parameter. We investigate the point process η on [0,∞) of near-record values from an integer-valued, independent and identically distributed sequence, showing that it is a Bernoulli cluster process. We derive the probability generating functional of η and formulas for the expectation, variance and covariance of the counting variables η(A),A⊂[0,∞). We also derive the strong convergence and asymptotic normality of η([0,n]), as n→∞, under mild regularity conditions on the distribution of the observations. For heavy-tailed distributions, with square-summable hazard rates, we prove that η([0,n]) grows to a finite random limit and compute its probability generating function. We present examples of the application of our results to particular distributions, covering a wide range of behaviours in terms of their right tails.

Idioma: Inglés

DOI: 10.3390/math10142442

Año: 2022

Publicado en: Mathematics 10, 14 (2022), 2442
ISSN: 2227-7390

Factor impacto JCR: 2.4 (2022)

Categ. JCR: MATHEMATICS rank: 23 / 329 = 0.07 (2022) - Q1 - T1

Factor impacto CITESCORE: 3.5 - Engineering (Q2) - Mathematics (Q1) - Computer Science (Q2)

Factor impacto SCIMAGO: 0.446 - Computer Science (miscellaneous) (Q2) - Mathematics (miscellaneous) (Q2) - Engineering (miscellaneous) (Q2)

Tipo y forma: Artículo (Versión definitiva)