Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

Shuyang Bai ()
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Shuyang Bai: University of Georgia

Journal of Theoretical Probability, 2022, vol. 35, issue 2, 917-948

Abstract: Abstract 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.

Keywords: Limit theorem; Long-range dependence; Infinite ergodic theory; Multiple stochastic integral; 60F17 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01090-9

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