On geometric records: rate of appearance and magnitude
Raúl Gouet, Francisco Javier López, Gerardo Sanz · Journal of Statistical Mechanics Theory and Experiment · 2012
We study the long-term behavior of geometric records from a sequence { X n } n ≥ 1 of independent, nonnegative, random observations, with common continuous distribution function F . Given a parameter k > 1, the n th observation X n is a geometric record if X n > k max{ X 1 ,..., X n − 1 }, that is, if X n is k times greater than all preceding observations. This concept was introduced by Eliazar in 2005 ( Physica A 348 181), where the question of waiting times was addressed. We consider the number N n of geometric records among X 1 ,..., X n , and show that N n increases to a finite random limit , for very light-tailed F . For medium and heavy-tailed F , we prove that N n diverges to infinity, establish its growth rate and give conditions for asymptotic normality. We also analyze the magnitude of geometric records, pointing out an unexpected relationship with models of paralyzable counters in particle physics. Our results are presented in a discrete-time setting but we show how they can be translated into continuous time. Examples of applications to common families of distributions, such as Fréchet systems, are also provided.