Tokyo Journal of Mathematics

Classes of Infinitely Divisible Distributions on $\mathbf{R}^d$ Related to the Class of Selfdecomposable Distributions

Makoto MAEJIMA, Muneya MATSUI, and Mayo SUZUKI

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This paper studies new classes of infinitely divisible distributions on $\mathbf{R}^d$. Firstly, the connecting classes with a continuous parameter between the Jurek class and the class of selfdecomposable distributions are revisited. Secondly, the range of the parameter is extended to construct new classes and characterizations in terms of stochastic integrals with respect to Lévy processes are given. Finally, the nested subclasses of those classes are discussed and characterized in two ways: One is by stochastic integral representations and another is in terms of Lévy measures.

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Tokyo J. Math., Volume 33, Number 2 (2010), 453-486.

First available in Project Euclid: 31 January 2011

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MAEJIMA, Makoto; MATSUI, Muneya; SUZUKI, Mayo. Classes of Infinitely Divisible Distributions on $\mathbf{R}^d$ Related to the Class of Selfdecomposable Distributions. Tokyo J. Math. 33 (2010), no. 2, 453--486. doi:10.3836/tjm/1296483482.

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  • O. E. Barndorff-Nielsen, M. Maejima and K. Sato, Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations, Bernoulli 12 (2006), 1–33.
  • Z. J. Jurek, The class $L_m(Q)$ of probability measures on Banach spaces, Bull. Polish Acad. Sci. Math. 31 (1983), 51–62.
  • Z.J. Jurek, Relations between the s–selfdecomposable and selfdecomposable measures, Ann. Probab. 13 (1985), 592–608.
  • Z. J. Jurek, Random integral representations for classes of limit distributions similar to Levy class $L_0$. Probab. Th. Rel. Fields 78 (1988), 473–490.
  • Z. J. Jurek, The random integral representation hypothesis revisited : new class of s-selfdecomposable laws, in: Abstract and Applied Analysis ; Proc. Intern. Conf. Hanoi, 2002, World Scientific (2004), 495–514.
  • T. A. O'Connor, Infinitely divisible distributions similar to class L distributions, Z.W. 50 (1979), 265–271.
  • J. Rosiński, Tempering stable processes, Stoch. Proc. Appl. 117 (2007), 677–707.
  • K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999.
  • K. Sato, Additive processes and stochastic integrals, Illinois J. Math. 50 (2006a) (Doob Volume), 825–851.
  • K. Sato, Two families of improper stochastic integrals with respect to Lévy processes, ALEA Lat. Am. J. Probab. Math. Statist. 1 (2006b), 47–87.
  • S. J. Wolfe, On a continuous analogue of the stochastic difference equation $X_n = \rho X_{n-1}+ B_n$, Stoch. Proc. Appl. 12 (1982), 301–312.