Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows. (English) Zbl 1381.60081

Jung, Paul; Owada, Takashi; Samorodnitsky, Gennady

Ann. Probab. 45, No. 4, 2087-2130 (2017).

Authors’ 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 due to T. Owada and G. Samorodnitsky [Ann. Probab. 43, No. 1, 240–285 (2015; Zbl 1320.60090)] 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 X. Chen [Probab. Theory Relat. Fields 116, No. 1, 89–123 (2000; Zbl 0953.60008)].

Reviewer: Ivan Podvigin (Novosibirsk)

MSC:

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

Keywords:

infinitely divisible process; conservative flow; Harris recurrent Markov chain; functional central limit theorem; self-similar process; pointwise dual ergodicity; Darling-Kac theorem; fractional stable motion

Citations:

Zbl 1320.60090; Zbl 0953.60008