Weak convergence towards two independent Gaussian processes from a unique Poisson process

• Autores: Xavier Bardina i Simorra , David Bascompte Viladrich
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 61, Fasc. 2, 2010, págs. 191-204
• Idioma: español
• DOI: 10.1007/bf03191241
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• Resumen

  • We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes.

    As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and sub-fractional Brownian motion.

• Referencias bibliográficas
  • X. Bardina, The complex Brownian motion as a weak limit of processes constructed from a Poisson process, Stochastic Analysis and Related Topics,...
  • X. Bardina and C. Florit, Approximation in law to the d-parameter fractional Brownian sheet based on the functional invariance principle,...
  • T. Bojdecki, L.G. Gorostiza, and A. Talarczyk, Subfractional Brownian motion and its relation to occupation times, Statist. Probab. Lett....
  • T. Bojdecki, L.G. Gorostiza, and A. Talarczyk, Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to...
  • P. Billingsley, Convergence of Probability Measures, John Wiley & Sons Inc., New York, 1968.
  • R. Delgado and M. Jolis, Weak approximation for a class of Gaussian processes, J. Appl. Probab. 37 (2000), 400–407.
  • L. Decreusefond and A.S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1999), 177–214.
  • M. Kac, A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math. 4 (1974), 497–509.
  • P. Lei and D. Nualart, A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), 619–624.
  • J. Ruiz de Chávez and C. Tudor, A decomposition of subfractional Brownian motion, Math. Rep. (Bucur.) 61 (2009), 67–74.
  • D.W. Stroock, Lectures on Topics in Stochastic Differential Equations, Tata Institute of Fundamental Research Lectures on Mathematics and...
  • C. Tudor, Some properties of the subfractional Brownian motion, Stochastics 79 (2007), 431–448.
  • C. Tudor, Inner product spaces of integrands associated to subfractional Brownian motion, Statist. Probab. Lett. 78 (2008), 2201–2209.