On geometric records: rate of appearance and magnitude

Abstract

We study the long-term behavior of geometric records from a sequence{Xn}n ≥ 1 of independent, nonnegative, random observations, with common continuous distribution functionF. Given aparameter k > 1, thenth observationXn is a geometricrecord if Xn > kmax{X1,..., Xn − 1}, that is, if Xn 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 numberNn of geometricrecords among X1,..., Xn,and show that Nn increases to a finite random limit , for very light-tailed F. Formedium and heavy-tailed F, we prove that Nn diverges to infinity, establish its growth rate and give conditions for asymptotic normality.We also analyze the magnitude of geometric records, pointing out an unexpectedrelationship with models of paralyzable counters in particle physics. Our results arepresented in a discrete-time setting but we show how they can be translated intocontinuous time. Examples of applications to common families of distributions, such asFréchet systems, are also provided.