## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 4 (2017), 2087-2130.

### Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

Paul Jung, Takashi Owada, and Gennady Samorodnitsky

#### 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].

#### Article information

**Source**

Ann. Probab., Volume 45, Number 4 (2017), 2087-2130.

**Dates**

Received: April 2015

Revised: December 2015

First available in Project Euclid: 11 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1502438423

**Digital Object Identifier**

doi:10.1214/16-AOP1107

**Mathematical Reviews number (MathSciNet)**

MR3693958

**Zentralblatt MATH identifier**

1381.60081

**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 Harris recurrent Markov chain functional central limit theorem self-similar process pointwise dual ergodicity Darling–Kac theorem fractional stable motion

#### Citation

Jung, Paul; Owada, Takashi; Samorodnitsky, Gennady. Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 45 (2017), no. 4, 2087--2130. doi:10.1214/16-AOP1107. https://projecteuclid.org/euclid.aop/1502438423