A functional non-central limit theorem for multiple-stable processes with long-range dependence

Abstract

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets. The joint moments of the local times are computed, which play the key in the proof and are also of independent interest.

arXiv comment40 pages

References in corpus (7)

• Operator renewal theory and mixing rates for dynamical systems with infinite measure
• Generalized Hermite processes, discrete chaos and limit theorems
• Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions
• Stochastic integral representations and classification of sum- and max-infinitely divisible processes
• Time-changed extremal process as a random sup measure
• Extremal theory for long range dependent infinitely divisible processes
• Representations of Hermite processes using local time of intersecting stationary stable regenerative sets