Renewals for exponentially increasing lifetimes, with an application to digital search trees

TL;DR

This paper investigates the behavior of renewal processes with exponentially increasing lifetimes, explaining the asymptotics of insertion depth in certain random trees and providing limit laws and convergence rates.

Contribution

It introduces a probabilistic framework for renewal processes with exponential lifetime growth and applies it to analyze random tree insertion depths.

Findings

Distributional fluctuations in renewal counts for exponential lifetimes

Representation of limit laws along subsequences

Rates of convergence for the renewal process

Abstract

We show that the number of renewals up to time $t$ exhibits distributional fluctuations as $t\to\infty$ if the underlying lifetimes increase at an exponential rate in a distributional sense. This provides a probabilistic explanation for the asymptotics of insertion depth in random trees generated by a bit-comparison strategy from uniform input; we also obtain a representation for the resulting family of limit laws along subsequences. Our approach can also be used to obtain rates of convergence.

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