FUNCTIONAL CENTRAL LIMIT THEOREM FOR A CLASS OF NEGATIVELY DEPENDENT HEAVY-TAILED STATIONARY INFINITELY DIVISIBLE PROCESSES GENERATED BY CONSERVATIVE FLOWS

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].