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Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

Journal of Theoretical Probability volume 35, pages 917–948 (2022)Cite this article

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 and 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 slowly enough.

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Acknowledgements

The author would like to thank Takashi Owada and Yizao Wang for helpful discussions. The author would like to thank the anonymous referees for their careful reading and helpful suggestions which have lead to substantial improvements of the paper.

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  1. Department of Statistics, University of Georgia, 310 Herty Drive, Athens, GA, 30602, USA

    Shuyang Bai

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  1. Shuyang Bai
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Correspondence to Shuyang Bai.

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Bai, S. Limit Theorems for Conservative Flows on Multiple Stochastic Integrals. J Theor Probab 35, 917–948 (2022). https://doi.org/10.1007/s10959-021-01090-9

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  • DOI: https://doi.org/10.1007/s10959-021-01090-9

Keywords

  • Limit theorem
  • Long-range dependence
  • Infinite ergodic theory
  • Multiple stochastic integral

Mathematics Subject Classification (2020)

  • 60F17