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.
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机译：我们显示，如果基础寿命在分布意义上以指数速率增长，则直到$ t $为止的续签次数都显示出$ t \ to \ infty $的分布波动。这为通过位比较策略根据统一输入生成的随机树中插入深度的渐近性提供了概率解释；我们还获得了伴随子序列的极限法则族的表示。我们的方法也可以用来获得收敛速度。
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