Abstract
We consider the sample average of a centered random walk in 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.
Funding Statement
The research of AB and BZ is partially supported by Dutch Science foundation NWO VICI grant # 639.033.413. The research of AB and ZP is partially supported by Polish National Science Centre Grant # 2018/29/B/ST1/00756 (2019-2022).
Acknowledgement
The authors are thankful to Guido Janssen for an analytic computation which led to a correct guess of the proper normalization of . The authors are thankful to a referee for valuable suggestions which improved the quality of the exposition.
Citation
Ayan Bhattacharya. Zbigniew Palmowski. Bert Zwart. "Persistence of heavy-tailed sample averages: principle of infinitely many big jumps." Electron. J. Probab. 27 1 - 25, 2022. https://doi.org/10.1214/22-EJP774
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