We show that the number of renewals up to time t exhibits distributional fluctuations as t→∞ 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.
@article{1174323260,
author = {Dennert, Florian and Gr\"ubel, Rudolf},
title = {Renewals for exponentially increasing lifetimes, with an application to digital search trees},
journal = {Ann. Appl. Probab.},
volume = {17},
number = {1},
year = {2007},
pages = { 676-687},
language = {en},
url = {http://dml.mathdoc.fr/item/1174323260}
}
Dennert, Florian; Grübel, Rudolf. Renewals for exponentially increasing lifetimes, with an application to digital search trees. Ann. Appl. Probab., Tome 17 (2007) no. 1, pp. 676-687. http://gdmltest.u-ga.fr/item/1174323260/