Abstract
We investigate a family of discrete-time stationary processes defined by multiple stable integrals and renewal processes with infinite means. The model may exhibit behaviors of short-range or long-range dependence, respectively, depending on the parameters. The main contribution is to establish a phase transition in terms of the tail processes that characterize local clustering of extremes. Moreover, in the short-range dependence regime, the model provides an example where the extremal index is different from the candidate extremal index.
Funding Statement
The second author was supported in part by Army Research Office, USA (W911NF-20-1-0139).
Acknowledgments
The authors are grateful to Gennady Samorodnitsky for suggesting to investigate the tail processes for stable-regenerative multiple-stable processes and for several helpful discussions, and to Rafał Kulik for very detailed explanations regarding tail processes and extremal indices, a very careful reading of a primitive version of the paper, and stimulating discussions. The authors would also like to thank Olivier Wintenberger for pointing out the reference Smith (1988). The authors thank two anonymous referees for careful reading and constructive comments.
Citation
Shuyang Bai. Yizao Wang. "Tail processes for stable-regenerative multiple-stable model." Bernoulli 29 (4) 3255 - 3279, November 2023. https://doi.org/10.3150/22-BEJ1582
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