Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

TL;DR

This paper establishes limit theorems for stationary sequences derived from multiple stochastic integrals in infinite-measure dynamical systems, revealing conditions for convergence to Gaussian or non-Gaussian processes based on covariance decay.

Contribution

It introduces new limit theorems for partial sums of such sequences, linking covariance decay rates to convergence to Brownian motion or fractional processes.

Findings

Central limit theorem with Brownian motion for fast covariance decay

Non-central limit theorems with fractional Brownian motion or Rosenblatt process for slow decay

Covariance decay rate determines the type of limiting process

Abstract

We consider a stationary sequence $(X_n)$ constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian, infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter $\beta\in(0,1)$ quantifying the conservativity of the system. This parameter $\beta$ together with the order of the integral determines the decay rate of the covariance of $(X_n)$. The goal of the paper is to establish limit theorems for the partial sum process of $(X_n)$. We obtain a central limit theorem with Brownian motion as limit when the covariance decays fast enough, as well as a non-central limit theorem with fractional Brownian motion or Rosenblatt process as limit when the covariance decays slow enough.