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Asymptotic behavior of the weak approximation to a class of Gaussian processes

Part of: Limit theorems Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  16 September 2021

Hui Jiang  and
Qingshan Yang
Hui Jiang*
Affiliation:
Nanjing University of Aeronautics and Astronautics
Qingshan Yang*
Affiliation:
Northeast Normal University
*
*Postal address: Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, P.R. China
**Postal address: School of Mathematics and Statistics, Northeast Normal University, Changchun, P.R. China. Email: yangqr66@gmail.com
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Abstract

We study, under mild conditions, the weak approximation constructed from a standard Poisson process for a class of Gaussian processes, and establish its sample path moderate deviations. The techniques consist of a good asymptotic exponential approximation in moderate deviations, the Besov–Lèvy modulus embedding, and an exponential martingale technique. Moreover, our results are applied to the weak approximations associated with the moving average of Brownian motion, fractional Brownian motion, and an Ornstein–Uhlenbeck process.

Keywords

Exponential martingaleGaussian processmoderate deviationsPoisson processweak approximation

MSC classification

Primary: 60F10: Large deviations
Secondary: 60F17: Functional limit theorems; invariance principles 60B10: Convergence of probability measures
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 693 - 707
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Applebaum, D. (2009). Lèvy Processes and Stochastic Calculus. Cambridge University Press.CrossRefGoogle Scholar
Bardina, X., Binotto, G. and Rovira, C. (2016). The complex Brownian motion as a strong limit of processes constructed from a Poisson process. J. Math. Anal. Appl. 444, 700–720.CrossRefGoogle Scholar
Bardina, X. and Jolis, M. (2000). Weak approximation of the Brownian sheet from a Poisson process in the plane. Bernoulli 6, 653–665.10.2307/3318512CrossRefGoogle Scholar
Bardina, X., Jolis, M. and Quer-Sardanyons, L. (2010). Weak convergence for the stochastic heat equation driven by Gaussian white noise. Electron. J. Prob. 15, 1267–1295.CrossRefGoogle Scholar
Bardina, X., Nourdin, I., Rovira, C. and Tindel, S. (2010). Weak approximation of a fractional SDE. Stoch. Proc. Appl. 120, 39–65.10.1016/j.spa.2009.10.008CrossRefGoogle Scholar
Bouè, M. and Dupuis, P. (1998). A variational representation for certain functionals of Brownian motion. Ann. Prob. 26, 1641–1659.CrossRefGoogle Scholar
Budhiraja, A., Dupuis, P. and Maroulas, V. (2011). Variational representation for continuous time processes. Ann. Inst. H. Poincarè Prob. Statist. 47, 725–747.Google Scholar
Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214.CrossRefGoogle Scholar
Delgado, R. and Jolis, M. (2000). Weak approximation for a class of Gaussian processes. J. Appl. Prob. 37, 400–407.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, Berlin.Google Scholar
Deuschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, London.Google Scholar
Deya, A., Jolis, M. and Quer-Sardanyons, L. (2013). The Stratonovich heat equation: a continuity result and weak approximations. Electron. J. Prob. 39, 1267–1295.Google Scholar
Feng, J. and Kurtz, T. G. (2006). Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs, 131, American Mathematical Society, Providence, RI.Google Scholar
Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Path: Theory and Applications. Cambridge University Press.CrossRefGoogle Scholar
Li, Y. Q. and Dai, H. S. (2011). Approximations of fractional Brownian motion. Bernoulli 17, 1195–1216.CrossRefGoogle Scholar
Nualart, D. (2006). The Malliavin Calculus and Related Topics. Springer, Berlin.Google Scholar
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Application. Gordon and Breach Science Publishers, Philadelphia, PA.Google Scholar
Stroock, D. W. (1982). Lectures on Topics in Stochastic Differential Equations. Springer, Berlin.Google Scholar
Wang, Z., Yan, L. T. and Yu, X. Y. (2013). Weak approximation of the fractional Brownian sheet from random walks. Electron. Commun. Prob. 18, 1–13.CrossRefGoogle Scholar
Zabczyk, J. and Da Prato, G. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press.Google Scholar