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 Probability Surveys > Vol. 16 (2019) open journal systems 

Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences

Lauri Viitasaari, University of Helsinki

The quadratic variation of Gaussian processes plays an important role in both stochastic analysis and in applications such as estimation of model parameters, and for this reason the topic has been extensively studied in the literature. In this article we study the convergence of quadratic sums of general Gaussian sequences. We provide necessary and sufficient conditions for different types of convergence including convergence in probability, almost sure convergence, $L^{p}$-convergence as well as weak convergence. We use a practical and simple approach which simplifies the existing methodology considerably. As an application, we show how convergence of the quadratic variation of a given process can be obtained by an appropriate choice of the underlying sequence.

AMS 2000 subject classifications: 60G15; 60F05; 60F15; 60F25

Keywords: Quadratic variations; Gaussian vectors; Gaussian processes; convergence in probability; strong convergence; convergence in $L^p$; central limit theorem

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Viitasaari, Lauri, Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences, Probability Surveys, 16, (2019), 62-98 (electronic). DOI: 10.1214/15-PS267.


[1]    X. Bardina and K. Es-Sebaiy. An extension of bifractional Brownian motion. Communications on Stochastic Analysis, 5(2):333–340, 2011. MR2814481

[2]    O.E. Barndoff-Nielsen, J.M. Corcuera, and M. Podolskij. Power variation for Gaussian processes with stationary increments. Stoch. Proc. Appl., 119(6):1845–1865, 2009. MR2519347

[3]    F. Baudoin and M. Hairer. A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Relat. Fields, 139:373–395, 2007. MR2322701

[4]    G. Baxter. A strong limit theorem for Gaussian processes. Proc. Amer. Soc., 7:522–527, 1997. MR0090920

[5]    A. Begyn. Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Prob., 10(20):691–717, 2005. MR2164027

[6]    A. Begyn. Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli, 13(3):712–753, 2007. MR2348748

[7]    A. Begyn. Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoch. Proc. Appl., 117:1848–1869, 2007. MR2437732

[8]    A. Benassi, S. Cohen, and J. Istas. Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett., 39:337–345, 1998. MR1646220

[9]    A. Benassi, S. Cohen, J. Istas, and S. Jaffard. Identification of filtered white noises. Stoch. Prob. Appl., 75:31–49, 1998. MR1629014

[10]    T. Bojdecki, L.G. Gorostiza, and A. Talarczyk. Sub-fractional Brownian motion and its relation to occupation time. Stat. Probab. Lett., 69:405–419, 2004. MR2091760

[11]    J.-C. Breton and I. Nourdin. Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electron. Comm. Probab., 13:482–493, 2008. MR2447835

[12]    P. Breuer and P. Major. Central limit theorems for non-linear functionals of Gaussian fields. Journal of Multivariate Analysis, 13:425–441, 1983. MR0716933

[13]    J. Coeurjolly. Identification of the multifractional Brownian motion. Bernoulli, 11:987–1009, 2005. MR2188838

[14]    S. Cohen, X. Guyon, O. Perrin, and M. Pontier. Singularity functions for fractional processes: application to fractional Brownian sheet. Annales de l’Institut Henri Poincaré, 42(2):187–205, 2006. MR2199797

[15]    J.M. Corcuera, D. Nualart, and J.H.C. Woerner. Power variation of some integral fractional processes. Bernoulli, 12(4):713–735, 2006. MR2248234

[16]    W.F. De La Vega. On almost sure convergence of quadratic Brownian variation. The Annals of Probability, 2:551–552, 1974. MR0359029

[17]    R.M. Dudley. Sample functions of the gaussian process. The Annals of Probability, 1:66–103, 1973. MR0346884

[18]    E.G. Gladyshev. A new limit theorem for stochastic processes with Gaussian increments. Theor. Prob. Appl., 6(1):52–61, 1961. MR0145574

[19]    X. Guyon and J. Léon. Convergence en loi des h-variations d’un processus gaussien stationnaire sur r. Annales de l’Institut Henri Poincaré, 25(3):265–282, 1989. MR1023952

[20]    D. Hanson and F. Wright. A bound on tail probabilities for quadratic forms in indepedent random variables. Annals of Mathematical Statistics, 42:1079–1083, 1971. MR0279864

[21]    C. Houdré and J. Villa. An example of infinite dimensional quasi-helix. Contemporary Mathematics, Amer. Math. Soc., 336:195–201, 2003. MR2037165

[22]    J. Istas and G. Lang. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Annales de l’Institut Henri Poincaré, 33(4):407–436, 1997. MR1465796

[23]    J.-F. Coeurjolly. Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process., 4:199–227, 2001. MR1856174

[24]    R. Klein and E. Gine. On quadratic variations of processes with Gaussian increments. The Annals of Probability, 3(4):716–721, 1975. MR0378070

[25]    K. Kubilius and D. Melichov. On the convergence rates of Gladyshev’s Hurst index estimator. Nonlinear Anal. Model. Control, 15:445–450, 2010. MR2881900

[26]    C. Lacaux. Real harmonizable multifractional lévy motions. Annales de l’Institut Henri Poincaré, 40(3):259–277, 2004. MR2060453

[27]    S. Levental and R.V. Erickson. On almost sure convergence of the quadratic variation of Brownian motion. Stoch. Proc. Appl., 106(2):317–333, 2003. MR1989631

[28]    J. Levy-Vehel and R.F. Peltier. Multifractional Brownian motion: definition and preliminary results. Rapport de recherche de l’INRIA, 2645, 1995.

[29]    R. Malukas. Limit theorems for a quadratic variation of Gaussian processes. Nonlinear Analysis: Modelling and Control, 16(4):435–452, 2011. MR2885721

[30]    R. Norvaiśa. A complement to Gladyshev’s theorem. Lith. Math. J., 51:26–35, 2011. MR2784375

[31]    I. Nourdin, D. Nualart, and C. Tudor. Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Annales de l’Institut Henri Poincaré, 46(4):1055–1079, 2010. MR2744886

[32]    I. Nourdin and G. Peccati. Normal Approximations Using Malliavin Calculus: from Stein’s Method to Universality. Cambridge University Press, 2012. MR2962301

[33]    I. Nourdin and G. Poly. Convergence in law in the second Wiener/Wigner chaos. Electron. Comm. Probab., 17(36). DOI:10.1214/ECP.v17-2023, 2012. MR2970700

[34]    I. Nourdin and G. Poly. Erratum: Convergence in law in the second Wiener/Wigner chaos. Electron. Comm. Probab., 17(54). DOI:10.1214/ ECP.v17-2383, 2012. MR2999982

[35]    I. Nourdin and A. Réveillac. Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case h = 14. The Annals of Probability, 37(6):2200–2230, 2009. MR2573556

[36]    D. Nualart. The Malliavin Calculus and Related Topics. Probability and Its Applications. Springer, 2006. MR2200233

[37]    D. Nualart and S. Ortiz-Latorre. Central limit theorems for multiple stochastic integrals and malliavin calculus. Stoch. Proc. Appl., 118(4):614–628, 2008. MR2394845

[38]    D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. The Annals of Probability, 33(1):177–193, 2005. MR2118863

[39]    M. Pakkanen. Limit theorems for power variations of ambit fields driven by white noise. Stoch. Proc. Appl., 124(5):1942–1973, 2014. MR3170230

[40]    M. Pakkanen and A. Réveillac. Functional limit theorems for generalized variations of the fractional Brownian sheet. Bernoulli, 22(3):1671–1708, 2016. MR3474829

[41]    G. Peccati and M. Taqqu. Wiener Chaos: Moments, Cumulants and Diagrams. A survey with computer implementation. Bocconi University Press, Springer, 2011. MR2791919

[42]    O. Perrin. Quadratic variation for Gaussian processes and application to time deformation. Stoch. Proc. Appl., 40(3):293–305, 1999. MR1700011

[43]    A. Réveillac. Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets. Stochastic Analysis and Applications, 27(1):51–73, 2009. MR2473140

[44]    A. Réveillac, M. Stauch, and C. Tudor. Hermite variations of the fractional Brownian sheet. Stoch. Dyn., DOI:10.1142/S0219493711500213, 2012. MR2926578

[45]    F. Russo and C. Tudor. On the bifractional Brownian motion. Stoch. Proc. Appl., 116(5):830–856, 2006. MR2218338

[46]    F. Russo and P. Vallois. Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics and Stochastics Reports, 70(1–2):1–40, 2000. MR1785063

[47]    T. Sottinen and L. Viitasaari. Stochastic analysis of Gaussian processes via Fredholm representation. International Journal of Stochastic Analysis, DOI:10.1155/2016/8694365, 2016. MR3536393

[48]    M. Taqqu. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab. Theory Relat. Fields, 31(4):287–302, 1975. MR0400329

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