Limit theorems for long-memory flows on Wiener chaos. (English) Zbl 1466.60072
Summary: We consider a long-memory stationary process, defined not through a moving average type structure, but by a flow generated by a measure-preserving transform and by a multiple Wiener-Itô integral. The flow is described using a notion of mixing for infinite-measure spaces introduced by K. Krickeberg [in: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Vol. II, Part 2: Contributions to probability theory. Berkeley-Los Angeles: University of California Press. 431–446 (1967; Zbl 0211.48503)]. Depending on the interplay between the spreading rate of the flow and the order of the multiple integral, one can recover known central or non-central limit theorems, and also obtain joint convergence of multiple integrals of different orders.
MSC:
| 60G10 | Stationary stochastic processes |
| 28D05 | Measure-preserving transformations |
| 60F05 | Central limit and other weak theorems |
| 60G15 | Gaussian processes |
| 60G18 | Self-similar stochastic processes |
Keywords:
conservative flows; ergodic theory; fourth moment theorem; long memory; long-range dependenceCitations:
Zbl 0211.48503Software:
longmemoReferences:
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