Joint Sum-and-Max Limit for a Class of Long-Range Dependent Processes with Heavy Tails

Shuyang Bai () and He Tang ()
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Shuyang Bai: University of Georgia
He Tang: University of Georgia

Journal of Theoretical Probability, 2024, vol. 37, issue 3, 1958-1987

Abstract: Abstract We consider a class of stationary processes exhibiting both long-range dependence and heavy tails. Separate limit theorems for sums and for extremes have been established recently in the literature with novel objects appearing in the limits. In this article, we establish the joint sum-and-max limit theorems for this class of processes. In the finite-variance case, the limit consists of two independent components: a fractional Brownian motion arising from the sum and a long-range dependent random sup measure arising from the maximum. In the infinite-variance case, we obtain in the limit two dependent components: a stable process and a random sup measure whose dependence structure is described through the local time and range of a stable subordinator. For establishing the limit theorem in the latter case, we also develop a joint convergence result for the local time and range of subordinators, which may be of independent interest.

Keywords: Asymptotic dependence; Infinitely divisible process; Long-range dependence; Stable subordinator; Weak convergence; Primary 60F17; Secondary 60G10 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01289-y

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