Cumulative record times in a Poisson process

TL;DR

This paper establishes a strong law of large numbers and a functional central limit theorem for record times and record durations in a Poisson process as time approaches infinity.

Contribution

It provides new asymptotic results for the distribution of record times and durations in Poisson processes, extending understanding of their long-term behavior.

Findings

Strong law of large numbers for record counts

Functional central limit theorem for record durations

Asymptotic behavior characterized as t→∞

Abstract

We obtain a strong law of large numbers and a functional central limit theorem, as $t\to\infty$, for the number of records up to time $t$ and the Lebesgue measure (length) of the subset of the time interval $[0,t]$ during which the Poisson process is in a record lifetime.