Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

Abstract

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.