# 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.

## Key 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.