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Micropulses and Different Types of Brownian Motion

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Matthieu Marouby
Matthieu Marouby*
Affiliation:
Université Paul Sabatier
*
∗ Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France. Email address: marouby@math.univ-toulouse.fr
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Abstract

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In this paper we study sums of micropulses that generate different kinds of processes. Fractional Brownian motion and bifractional Brownian motion are obtained as limit processes. Moreover, we not only prove the convergence of finite-dimensional laws but also, in some cases, convergence in distribution in the space of right-continuous functions with left limits. Finally, we obtain generalizations with multidimensional indices.

Keywords

Sums of micropulsesfractional Brownian motionmultifractional Brownian motionPoisson random measure

MSC classification

Secondary: 60G15: Gaussian processes 60G22: Fractional processes, including fractional Brownian motion 60G60: Random fields
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 792 - 810
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Biermé, H. and Estrade, A. (2006). Poisson random balls: self-similarity and x-ray images. Adv. Appl. Prob. 38, 853–872.Google Scholar
[2] Biermé, H., Estrade, A. and Kaj, I. (2010). {Self-similar random fields and rescaled random balls models}. J. Theoret. Prob. 23, 1110–1141.Google Scholar
[3] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
[4] Breton, J.-C. and Dombry, C. (2009). Rescaled weighted random ball models and stable self-similar random fields. Stoch. Process. Appl. 119, 3633–3652.Google Scholar
[5] Cioczek-Georges, R. and Mandelbrot, B. B. (1995). A class of micropulses and antipersistent fractional Brownian motion. Stoch. Process. Appl. 60, 1–18.Google Scholar
[6] Cioczek-Georges, R. and Mandelbrot, B. B. (1996). Alternative micropulses and fractional Brownian motion. Stoch. Process. Appl. 64, 143–152.Google Scholar
[7] Houdré, C. and Villa, J. (2003). An example of infinite dimensional quasi-helix. In Stochastic Models (Mexico City, 2002; Contemp. Math. 336), American Mathematical Society, Providence, RI, pp. 195–201.Google Scholar
[8] Kaj, I., Leskelä, L., Norros, I. and Schmidt, V. (2007). Scaling limits for random fields with long-range dependence. Ann. Prob. 35, 528–550.Google Scholar
[9] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
[10] Kingman, J. F. C. (1993). Poisson Processes (Oxford Stud. Prob. 3). Oxford University Press, New York.Google Scholar
[11] Kolmogorov, A. (1940). {Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum}. C. R. (Doklady) Acad. URSS (NS) 26, 115–118.Google Scholar
[12] Mandelbrot, B. B. and Van Ness, J. W. (1968). {Fractional Brownian motions, fractional noises and applications}. SIAM Rev. 10, 422–437.Google Scholar
[13] Marouby, M. (2010). Trois études de processus fractionnaires. , Université Paul Sabatier. Available at http://thesesups.ups-tlse.fr/946/1/Marouby_Matthieu.pdf.Google Scholar
[14] Tudor, C. A. and Xiao, Y. (2007). Sample path properties of bifractional Brownian motion. Bernoulli 13, 1023–1052.Google Scholar