Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-20T00:47:33.292Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak approximation for a class of Gaussian processes

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

Published online by Cambridge University Press:  14 July 2016

Rosario Delgado  and
Maria Jolis
Rosario Delgado*
Affiliation:
Universitat Autònoma de Barcelona
Maria Jolis*
Affiliation:
Universitat Autònoma de Barcelona
*
∗Postal address: Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
∗Postal address: Departament de Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Keywords

Weak convergenceGaussian processPoisson processfractional Brownian motion

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60B10: Convergence of probability measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 400 - 407
Copyright
Copyright © by the Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Partly supported by grants PB96 0088 and PB96 1182, Dirección General de Enseñanza Superior, and 19955GR593, CIRIT.

References

Billingsley, P. (1968). Convergence of Probability measures. John Wiley. New York.Google Scholar
Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Analysis 10, 177–214.Google Scholar
Hida, T., and Hitsuda, M. (1976). Gaussian Processes (Translations of Mathematical Monographs 120). American Mathematical Society, Providence, RI.Google Scholar
Mandelbrot, B. B., and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.CrossRefGoogle Scholar
Stroock, D. W. (1982). Topics in Stochastic Differential Equations. Tata Institute of Fundamental Research & Springer, Berlin.Google Scholar