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

TL;DR

This paper investigates conditions under which record values are finite, especially for iid variables with finite endpoints, and develops asymptotic theories for the sequence of hitting times using negative binomial and multinomial models.

Contribution

It provides necessary and sufficient conditions for the finiteness of record values and introduces asymptotic characterizations of hitting times using negative binomial and multinomial distributions.

Findings

Strong upper record values are finite iff the upper endpoint is finite and an atom of the distribution.

Asymptotic behavior of hitting times characterized by negative binomial and multinomial models.

Confidence bounds and comparisons for hitting times are derived.

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 \textsl{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 provide. The example of a binomial random variable is given