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Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

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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References

  1. Aaronson, J.: An Introduction to Infinite Ergodic Theory. Number 50. American Mathematical Society, Providence (1997)

    Book  Google Scholar 

  2. Bai, S.: Representations of Hermite processes using local time of intersecting stationary stable regenerative sets. J. Appl. Probab. (2019). arXiv preprint arXiv:1910.07120

  3. Bai, S., Taqqu, M.S.: Limit theorems for long-memory flows on Wiener chaos. Bernoulli 26(2), 1473–1503 (2020)

    MathSciNet  Article  Google Scholar 

  4. Bai, S., Owada, T., Wang, Y.: A functional non-central limit theorem for multiple-stable processes with long-range dependence. Stoch. Process. Appl. 130(9), 5768–5801 (2020)

    MathSciNet  Article  Google Scholar 

  5. Bingham, N., Goldie, C., Teugels, J.: Regular Variation. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  6. Chen, Z., Samorodnitsky, G.: Extremal clustering under moderate long range dependence and moderately heavy tails. arXiv preprint arXiv:2003.05038 (2020a)

  7. Chen, Z., Samorodnitsky, G.: Extreme value theory for long-range-dependent stable random fields. J. Theor. Probab. 33(4), 1894–1918 (2020b)

    MathSciNet  Article  Google Scholar 

  8. Dobrushin, R., Major, P.: Non-central limit theorems for non-linear functional of Gaussian fields. Probab. Theory Relat. Fields 50(1), 27–52 (1979)

    MATH  Google Scholar 

  9. Giraitis, L., Koul, H., Surgailis, D.: Large Sample Inference for Long Memory Processes. World Scientific Publishing Company Incorporated, Hackensack (2012)

    Book  Google Scholar 

  10. Gouëzel, S.: Correlation asymptotics from large deviations in dynamical systems with infinite measure. In: Colloquium Mathematicum, vol. 125, pp. 193–212. Instytut Matematyczny Polskiej Akademii Nauk (2011)

  11. Hajian, A.B., Kakutani, S.: Weakly wandering sets and invariant measures. Trans. Am. Math. Soc. 110(1), 136–151 (1964)

    MathSciNet  Article  Google Scholar 

  12. Itô, K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3(1), 157–169 (1951)

    MathSciNet  Article  Google Scholar 

  13. Janson, S.: Gaussian Hilbert Spaces, vol. 129. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  14. Jung, P., Owada, T., Samorodnitsky, G.: Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 45(4), 2087–2130 (2017)

    MathSciNet  Article  Google Scholar 

  15. Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2002)

    Book  Google Scholar 

  16. Kallenberg, O.: Random Measures Theory and Applications. Springer, Berlin (2017)

    Book  Google Scholar 

  17. Kesseböhmer, M., Slassi, M.: Limit laws for distorted critical return time processes in infinite ergodic theory. Stoch. Dyn. 7(01), 103–121 (2007)

    MathSciNet  Article  Google Scholar 

  18. Krickeberg, K.: Strong mixing properties of Markov chains with infinite invariant measure. In: Proceedings Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), vol. 2, pp. 431–446 (1967)

  19. Lacaux, C., Samorodnitsky, G.: Time-changed extremal process as a random sup measure. Bernoulli 22(4), 1979–2000 (2016)

    MathSciNet  Article  Google Scholar 

  20. Melbourne, I., Terhesiu, D.: Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 189(1), 61–110 (2012)

    MathSciNet  Article  Google Scholar 

  21. Owada, T.: Limit theory for the sample autocovariance for heavy-tailed stationary infinitely divisible processes generated by conservative flows. J. Theor. Probab. 29(1), 63–95 (2016)

    MathSciNet  Article  Google Scholar 

  22. Owada, T., Samorodnitsky, G.: Maxima of long memory stationary symmetric \(alpha\)-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21(3), 1575–1599 (2015a)

    MathSciNet  Article  Google Scholar 

  23. Owada, T., Samorodnitsky, G.: Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43(1), 240–285 (2015b)

    MathSciNet  Article  Google Scholar 

  24. Peccati, G., Taqqu, M.: Wiener Chaos: Moments, Cumulants and Diagrams: a Survey with Computer Implementation. Springer, Berlin (2011)

    Book  Google Scholar 

  25. Pipiras, V., Taqqu, M.: Long-Range Dependence and Self-Similarity, vol. 45. Cambridge University Press, Cambridge (2017)

    Book  Google Scholar 

  26. Rosinski, J.: On series representations of infinitely divisible random vectors. Ann Probab 405–430 (1990)

  27. Rosinski, J., Samorodnitsky, G.: Product formula, tails and independence of multiple stable integrals. Adv. Stoch. Inequal. (Atlanta, GA, 1997) 234, 169–194 (1999)

    MathSciNet  Article  Google Scholar 

  28. Samorodnitsky, G.: Stochastic Processes and Long Range Dependence, vol. 26. Springer, Berlin (2016)

    Book  Google Scholar 

  29. Samorodnitsky, G., Szulga, J.: An asymptotic evaluation of the tail of a multiple symmetric \(\alpha \)-stable integral. Ann. Probab. 1503–1520 (1989)

  30. Samorodnitsky, G., Wang, Y.: Extremal theory for long range dependent infinitely divisible processes. Ann. Probab. 47(4), 2529–2562 (2019)

    MathSciNet  Article  Google Scholar 

  31. Schmüdgen, K.: The Moment Problem, vol. 9. Springer, Berlin (2017)

    Book  Google Scholar 

  32. Slud, E.: The moment problem for polynomial forms in normal random variables. Ann Probab. 2200–2214 (1993)

  33. Szulga, J.: Multiple stochastic integrals with respect to symmetric infinitely divisible random measures. Ann. Probab. 1145–1156 (1991)

  34. Taqqu, M.: Convergence of integrated processes of arbitrary Hermite rank. Probab. Theory Relat. Fields 50(1), 53–83 (1979)

    MathSciNet  MATH  Google Scholar 

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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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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