We find sharp asymptotics of the probability that the trajectory of a compound renewal process crosses (or does not cross) an arbitrary remote boundary. In particular, some limit theorems are obtained for the distribution of the maximum of the process in the domain of large deviations. We also give some applications to the classical ruin probability problem in insurance theory.

中文翻译：

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复合更新过程的边界交叉问题

我们发现复合更新过程的轨迹跨越（或不跨越）任意远程边界的概率的急剧渐近。特别地，对于过程的最大值在大偏差域中的分布，得到了一些极限定理。我们还对保险理论中的经典破产概率问题给出了一些应用。