The Annals of Probability

Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows

Takashi Owada and Gennady Samorodnitsky

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Abstract

We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying Lévy measures. The limit process is a new class of symmetric stable self-similar processes with stationary increments that coincides on a part of its parameter space with a previously described process. The normalizing sequence and the limiting process are determined by the ergodic-theoretical properties of the flow underlying the integral representation of the process. These properties can be interpreted as determining how long the memory of the stationary infinitely divisible process is. We also establish functional convergence, in a strong distributional sense, for conservative pointwise dual ergodic maps preserving an infinite measure.

Article information

Source
Ann. Probab., Volume 43, Number 1 (2015), 240-285.

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1415801557

Digital Object Identifier
doi:10.1214/13-AOP899

Mathematical Reviews number (MathSciNet)
MR3298473

Zentralblatt MATH identifier
1320.60090

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G18: Self-similar processes
Secondary: 37A40: Nonsingular (and infinite-measure preserving) transformations 60G52: Stable processes

Keywords
Infinitely divisible process conservative flow central limit theorem self-similar process pointwise dual ergodicity Darling–Kac theorem

Citation

Owada, Takashi; Samorodnitsky, Gennady. Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43 (2015), no. 1, 240--285. doi:10.1214/13-AOP899. https://projecteuclid.org/euclid.aop/1415801557


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