# Renewals for exponentially increasing lifetimes, with an application to digital search trees - Mathematics Probability

Abstract: We show that the number of renewals up to time $t$ exhibits distributionalfluctuations as $t\to\infty$ if the underlying lifetimes increase at anexponential rate in a distributional sense. This provides a probabilisticexplanation for the asymptotics of insertion depth in random trees generated bya bit-comparison strategy from uniform input; we also obtain a representationfor the resulting family of limit laws along subsequences. Our approach canalso be used to obtain rates of convergence.

Author: Florian Dennert, Rudolf Grübel

Source: https://arxiv.org/