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Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

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  • Takashi Owada

    (Cornell University)

Abstract

This study aim to develop limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process has heavy tail marginals and is generated by a conservative flow. Interestingly, the growth rate of the sample autocovariances is determined not only by heavy tailedness of the marginals but also by the memory length of the process. Although this feature was first observed by Resnick et al. (Stoch Process Appl 85:321–339, 2000) for some very specific processes, we will propose a more general framework from the viewpoint of infinite ergodic theory. Consequently, the asymptotics of the sample autocovariances can be more comprehensively discussed.

Suggested Citation

  • Takashi Owada, 2016. "Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows," Journal of Theoretical Probability, Springer, vol. 29(1), pages 63-95, March.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:1:d:10.1007_s10959-014-0565-9
    DOI: 10.1007/s10959-014-0565-9
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    References listed on IDEAS

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    1. Resnick, Sidney & Samorodnitsky, Gennady & Xue, Fang, 2000. "Growth rates of sample covariances of stationary symmetric [alpha]-stable processes associated with null recurrent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 321-339, February.
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    Cited by:

    1. Shuyang Bai, 2022. "Limit Theorems for Conservative Flows on Multiple Stochastic Integrals," Journal of Theoretical Probability, Springer, vol. 35(2), pages 917-948, June.
    2. Parthanil Roy, 2017. "Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 513-540, December.
    3. Zaoli Chen & Gennady Samorodnitsky, 2020. "Extreme Value Theory for Long-Range-Dependent Stable Random Fields," Journal of Theoretical Probability, Springer, vol. 33(4), pages 1894-1918, December.

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