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. (to be continued on page 2372).

Key words: record values; record time; endpoints of a cumulative distribution function; hitting times; central limit theorem; law of the iterated logarithm; Berry-Essen bound; band of consistency; confidence intervals