We consider the sample average of a centered random walk in Rd with regularly varying step size distribution. For the first exit time from a compact convex set A not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.

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Electronic Journal of Probability
Rare events: Asymptotics, Algorithms, Applications
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bhattacharya, A., Palmowski, Z., & Zwart, B. (2022). Persistence of heavy-tailed sample averages: principle of infinitely many big jumps. Electronic Journal of Probability, 27, 1–25. doi:10.1214/22-EJP774