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July 2017 Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows
Paul Jung, Takashi Owada, Gennady Samorodnitsky
Ann. Probab. 45(4): 2087-2130 (July 2017). DOI: 10.1214/16-AOP1107

Abstract

We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].

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Paul Jung. Takashi Owada. Gennady Samorodnitsky. "Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows." Ann. Probab. 45 (4) 2087 - 2130, July 2017. https://doi.org/10.1214/16-AOP1107

Information

Received: 1 April 2015; Revised: 1 December 2015; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1381.60081
MathSciNet: MR3693958
Digital Object Identifier: 10.1214/16-AOP1107

Subjects:
Primary: 60F17 , 60G18
Secondary: 37A40 , 60G52

Keywords: conservative flow , Darling–Kac theorem , Fractional stable motion , functional central limit theorem , Harris recurrent Markov chain , infinitely divisible process , pointwise dual ergodicity , self-similar process

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 4 • July 2017
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