Several forms of stochastic integral representations of gamma random variables and related topics

• Autorzy: Aoyama T. , Maejima M. , Ueda Y.
• Języki publikacji: EN

Abstrakty

EN

Gamma distributions can be characterized as the laws of stochastic integrals with respect to many different Lévy processes with different nonrandom integrands. A Lévy process corresponds to an infinitely divisible distribution. Therefore, many infinitely divisible distributions can yield a gamma distribution through stochastic integral mappings with different integrands. In this paper, we pick up several integrands which have appeared in characterizing well-studied classes of infinitely divisible distributions, and find inverse images of a gamma distribution through each stochastic integral mapping. As a by-product of our approach to stochastic integral representations of gamma random variables, we find a remarkable new general characterization of classes of infinitely divisible distributions, which were already considered by James et al. (2008) and Aoyama et al. (2010) in some special cases.

Słowa kluczowe

EN

infinitely divisible distribution; gamma distribution; stochastic integral representation; Lévy process

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 1
• Strony: 99--118
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Aoyama T.

• Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, 2641, Yamazaki, Noda, 278-8510, Japan

autor

Maejima M.

• Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

autor

Ueda Y.

• Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

Bibliografia

• [1] T. Aoyama, A. Lindner and M. Maejima, A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class, Electron. J. Probab. 15 (2010), pp. 1119-1142.
• [2] T. Aoyama and M. Maejima, Characterizations of subclasses of type G distributions on Rd by stochastic integral representations, Bernoulli 13 (2007), pp. 148-160.
• [3] T. Aoyama, M. Maejima and J. Rosiński, A subclass of type G selfdecomposable distributions on Rd, J. Theoret. Probab. 21 (2008), pp. 14-34.
• [4] O. E. Barndorff-Nielsen, M. Maejima and K. Sato, Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations, Bernoulli 12 (2006), pp. 1-33.
• [5] O. E. Barndorff-Nielsen, J. Rosinski and S. Thorbjørnsen, General ϒ-transformations, ALEA, Lat. Am. J. Probab. Math. Stat. 4 (2008), pp. 131-165.
• [6] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statist. 76, Springer, 1992.
• [7] L. F. James, B. Roynette and M. Yor, Generalized gamma convolution, Dirichlet means, Thorin measures, with explicit examples, Probability Surveys 5 (2008), pp. 346-415.
• [8] Z. J. Jurek, Relations between the s-selfdecomposable and selfdecomposable measures, Ann. Probab. 13 (1985), pp. 592-608.
• [9] Z. J. Jurek, Random integral representations for classes of limit distributions similar to Lévy class L0, Probab. Theory Related Fields 78 (1988), pp. 473-490.
• [10] Z. J. Jurek and W. Vervaat, An integral representation for selfdecomposable Banach space valued random variables, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 62 (1983), pp. 247-262.
• [11] M. Maejima, To which class do known infinitely divisible distributions belong? (Version 1), 2007. http://www.math.keio.ac.jp/˜maejima/
• [12] M. Maejima, M. Matsui and M. Suzuki, Classes of infinitely divisible distributions on Rd related to the class of selfdecomposable distributions, Tokyo J. Math. 33 (2010), pp. 453-486.
• [13] M. Maejima and G. Nakahara, A note on new classes of infinitely divisible distributions on Rd, Electron. Comm. Probab. 14 (2009), pp. 358-371.
• [14] M. Maejima, V. Pérez-Abreu and K. Sato, A class of multivariate infinitely divisible distributions related to arcsine density, Bernoulli (to appear).
• [15] M. Maejima and K. Sato, The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions, Probab. Theory Related Fields 145 (2009), pp. 119-142.
• [16] M. Maejima and Y. Ueda, Compositions of mappings of infinitely divisible distributions with applications to finding the limits of some nested subclasses, Electron. Comm. Probab. 15 (2010), pp. 227-239.
• [17] K. Sato, Subordination and self-decomposability, Statist. Probab. Lett. 54 (2001), pp. 317-324.
• [18] K. Sato, Two families of improper stochastic integrals with respect to Lévy processes, ALEA, Lat. Am. J. Probab. Math. Stat. 1 (2006), pp. 47-87.
• [19] K. Sato, Transformations of infinitely divisible distributions via improper stochastic integrals, ALEA, Lat. Am. J. Probab. Math. Stat. 3 (2007), pp. 67-110.
• [20] K. Sato and M. Yamazato, Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type, Stochastic Process. Appl. 17 (1984), pp. 73-100.
• [21] S. J. Wolfe, On a continuous analogue of the stochastic difference equation Xn = _Xn−1 +Bn, Stochastic Process. Appl. 12 (1982), pp. 301-312.
• [22] M. Yamazato, Unimodality of infinitely divisible distribution functions of class L, Ann. Probab. 6 (1978), pp. 523-531.