  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.
