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CLT for quadratic variation of Gaussian processes and its application to the estimation of the Orey index

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  • Kubilius, K.

Abstract

We give a two-dimensional central limit theorem (CLT) for the second-order quadratic variation of the centered Gaussian processes on [0,T]. Though the approach we use is well known in the literature, the conditions under which the CLT holds are usually based on differentiability of the corresponding covariance function. In our case, we replace differentiability conditions by the convergence of the scaled sums of the second-order moments. To illustrate the usefulness and easiness of use of the approach, we apply the obtained CLT to proving the asymptotic normality of the estimator of the Orey index of a subfractional Brownian motion.

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  • Kubilius, K., 2020. "CLT for quadratic variation of Gaussian processes and its application to the estimation of the Orey index," Statistics & Probability Letters, Elsevier, vol. 165(C).
    • Handle: RePEc:eee:stapro:v:165:y:2020:i:c:s0167715220301486
    DOI: 10.1016/j.spl.2020.108845
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    References listed on IDEAS

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    1. Jean-François Coeurjolly, 2001. "Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths," Statistical Inference for Stochastic Processes, Springer, vol. 4(2), pages 199-227, May.
    2. Kęstutis Kubilius & Dmitrij Melichov, 2016. "Exact Confidence Intervals of the Extended Orey Index for Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 785-804, September.
    3. 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.
    4. Benassi, Albert & Cohen, Serge & Istas, Jacques & Jaffard, Stéphane, 1998. "Identification of filtered white noises," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 31-49, June.
    5. 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.
    6. Kubilius, K. & Skorniakov, V., 2016. "On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 159-167.
    7. Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Moment bounds and central limit theorems for Gaussian subordinated arrays," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 457-473.
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    Cited by:

    1. Kęstutis Kubilius & Aidas Medžiūnas, 2020. "Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient," Mathematics, MDPI, vol. 9(1), pages 1-14, December.
    2. Yicun Li & Yuanyang Teng, 2022. "Estimation of the Hurst Parameter in Spot Volatility," Mathematics, MDPI, vol. 10(10), pages 1-26, May.

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