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Central limit theorem for functionals of a generalized self-similar Gaussian process

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  • Harnett, Daniel
  • Nualart, David

Abstract

We consider a class of self-similar, continuous Gaussian processes that do not necessarily have stationary increments. We prove a version of the Breuer–Major theorem for this class, that is, subject to conditions on the covariance function, a generic functional of the process increments converges in law to a Gaussian random variable. The proof is based on the Fourth Moment Theorem. We give examples of five non-stationary processes that satisfy these conditions.

Suggested Citation

  • Harnett, Daniel & Nualart, David, 2018. "Central limit theorem for functionals of a generalized self-similar Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 404-425.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:2:p:404-425
    DOI: 10.1016/j.spa.2017.04.014
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    References listed on IDEAS

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    1. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2004. "Sub-fractional Brownian motion and its relation to occupation times," RePAd Working Paper Series lrsp-TRS376, Département des sciences administratives, UQO.
    2. Bojdecki, Tomasz & Gorostiza, Luis G. & Talarczyk, Anna, 2004. "Sub-fractional Brownian motion and its relation to occupation times," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 405-419, October.
    3. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    4. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    5. Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
    6. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
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    Cited by:

    1. Nualart, David & Xu, Fangjun, 2019. "Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3981-4008.
    2. Daniel Harnett & Arturo Jaramillo & David Nualart, 2019. "Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1105-1144, September.

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