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Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

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  • Daniel Harnett

    (University of Wisconsin-Stevens Point)

  • Arturo Jaramillo

    (University of Kansas)

  • David Nualart

    (University of Kansas)

Abstract

We study the asymptotic behavior of the $$\nu $$ ν -symmetric Riemann sums for functionals of a self-similar centered Gaussian process X with increment exponent $$0 (2\ell +1)^{-1}$$ α > ( 2 ℓ + 1 ) - 1 , we prove that the convergence holds in probability.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0833-1
    DOI: 10.1007/s10959-018-0833-1
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    References listed on IDEAS

    as
    1. Ivan Nourdin & David Nualart, 2010. "Central Limit Theorems for Multiple Skorokhod Integrals," Journal of Theoretical Probability, Springer, vol. 23(1), pages 39-64, March.
    2. Harnett, Daniel & Nualart, David, 2012. "Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3460-3505.
    3. 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.
    4. 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.
    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.
    7. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    8. Swanson, Jason, 2011. "Fluctuations of the empirical quantiles of independent Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 479-514, March.
    9. Daniel Harnett & David Nualart, 2015. "On Simpson’s Rule and Fractional Brownian Motion with $$H = 1/10$$ H = 1 / 10," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1651-1688, December.
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