Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

We consider a stationary sequence $$(X_n)$$ ( 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 and infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) quantifying the conservativity of the system. This parameter $$\beta $$ β together with the order of the integral determines the decay rate of the covariance of $$(X_n)$$ ( X n ) . The goal of the paper is to establish limit theorems for the partial sum process of $$(X_n)$$ ( 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 slowly enough.