Bernoulli

  • Bernoulli
  • Volume 18, Number 4 (2012), 1099-1127.

A Ferguson–Klass–LePage series representation of multistable multifractional motions and related processes

R. Le Guével and J. Lévy Véhel

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Abstract

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. In (J. Theoret. Probab. 22 (2009) 375–401), a particular way of constructing such processes was investigated, leading in particular to multifractional multistable processes, which were built using sums over Poisson processes. We present here a different construction of these processes, based on the Ferguson–Klass–LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse Ornstein–Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also show that the processes defined here have the same finite dimensional distributions as the corresponding processes built in (J. Theoret. Probab. 22 (2009) 375–401). Finally, we display numerical experiments showing graphs of synthesized paths of such processes.

Article information

Source
Bernoulli, Volume 18, Number 4 (2012), 1099-1127.

Dates
First available in Project Euclid: 12 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1352727803

Digital Object Identifier
doi:10.3150/11-BEJ372

Mathematical Reviews number (MathSciNet)
MR2995788

Zentralblatt MATH identifier
1260.60096

Keywords
Ferguson–Klass–LePage series representation localisable processes multifractional processes stable processes

Citation

Guével, R. Le; Véhel, J. Lévy. A Ferguson–Klass–LePage series representation of multistable multifractional motions and related processes. Bernoulli 18 (2012), no. 4, 1099--1127. doi:10.3150/11-BEJ372. https://projecteuclid.org/euclid.bj/1352727803


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References

  • [1] Ayache, A. and Levy Vehel, J. (2000). The generalized multifractional Brownian motion. Stat. Inference Stoch. Process. 3 7–18.
  • [2] Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoam. 13 19–90.
  • [3] Bentkus, V., Juozulynas, A. and Paulauskas, V. (2001). Lévy–LePage series representation of stable vectors: Convergence in variation. J. Theoret. Probab. 14 949–978.
  • [4] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. New York: Springer.
  • [5] Falconer, K.J. (2002). Tangent fields and the local structure of random fields. J. Theoret. Probab. 15 731–750.
  • [6] Falconer, K.J. (2003). The local structure of random processes. J. London Math. Soc. (2) 67 657–672.
  • [7] Falconer, K.J., Le Guével, R. and Lévy Véhel, J. (2009). Localisable moving average stable and multistable processes. Stochastic Models 25 648–672.
  • [8] Falconer, K.J. and Lévy Véhel, J. (2008). Multifractional, multistable, and other processes with prescribed local form, J. Theoret. Probab., DOI 10.1007/s10959-008-0147-9.
  • [9] Ferguson, T.S. and Klass, M.J. (1972). A representation of independent increment processes without Gaussian components. Ann. Math. Statist. 43 1634–1643.
  • [10] Herbin, E. (2006). From $N$ parameter fractional Brownian motions to $N$ parameter multifractional Brownian motions. Rocky Mountain J. Math. 36 1249–1284.
  • [11] Kolmogoroff, A.N. (1940). Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 115–118.
  • [12] Le Guével, R. and Lévy Véhel, J. (2008). Incremental moments and Hölder exponents of multifractional multistable processes. Preprint. Available at http://arxiv.org/abs/0807.0764.
  • [13] Le Page, R. (1980). Multidimensionely divisible variables and processes. I. Stable case. Technical Report 292, Dept. Statistics, Stanford Univ.
  • [14] Le Page, R. (1981). Multidimensional infinitely divisible variables and processes. II. In Probability in Banach Spaces, III (Medford, MA, 1980). Lecture Notes in Math. 860 279–284. Berlin: Springer.
  • [15] Ledoux, M. and Talagrand, M. (1996). Probability in Banach Spaces. Berlin: Springer.
  • [16] Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [17] Peltier, R.F. and Lévy Véhel, J. (1995). Multifractional Brownian motion: Definition and preliminary results. Rapport de Recherche de L’INRIA, 2645. Available at http://hal.inria.fr/docs/00/07/40/45/PDF/RR-2645.pdf.
  • [18] Petrov, V.V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. New York: Oxford Univ. Press.
  • [19] Rosiński, J. (1990). On series representations of infinitely divisible random vectors. Ann. Probab. 18 405–430.
  • [20] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Stochastic Modeling. New York: Chapman & Hall.
  • [21] Stoev, S. and Taqqu, M.S. (2004). Stochastic properties of the linear multifractional stable motion. Adv. in Appl. Probab. 36 1085–1115.