Abstract
In this paper, we find and develop a stochastic integral representation for the class of strictly stable distributions. We establish an explicit relationship between stochastic integral and shot-noise series representations of strictly stable distributions, which shows that the class of distributions representable by stochastic integral is larger than the class representable by a shot-noise series. This inclusion is proper when the stability index \(\alpha \) is greater than 1. We also give an explicit description of distributions possessing both representations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aoyama, T., Lindner, A., Maejima, M.: A new family of mappings of infinitely divisible distributions related to the Goldie-Steutel-Bondesson class. Electron. J. Probab. 15, 1119–1142 (2010)
Aoyama, T., Maejima, M.: Characterizations of subclasses of type \(G\) distributions on \({\mathbb{R}}^d\) by stochastic integral representations. Bernoulli 13, 148–160 (2007)
Aoyama, T., Maejima, M., Rosiński, J.: A subclass of type \(G\) selfdecomposable distributions on \({\mathbb{R^d}}\). J. Theor. Probab. 21, 14–34 (2008)
Aoyama, T., Maejima, M., Ueda, Y.: Several forms of stochastic integral representations of gamma random variables and related topics. Probab. Math. Statist. 31, 99–118 (2011)
Barndorff-Nielsen, O.E., Maejima, M., Sato, K.: Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–33 (2006)
Barndorff-Nielsen, O.E., Rosinski, J., Thorbjørnsen, S.: General \(\Upsilon \)-transformations. ALEA Lat. Am. J. Probab. Math. Stat. 4, 131–165 (2008)
Ichifuji, K., Maejima, M., Ueda, Y.: Fixed points of mappings of infinitely divisible distributions on \({\mathbb{R^d}}\). Statist. Probab. Lett. 80, 1320–1328 (2010)
Jurek, Z.J.: Relations between the \(s\)-selfdecomposable and selfdecomposable measures. Ann. Probab. 13, 592–608 (1985)
Jurek, Z.J., Vervaat, W.: An integral representation for selfdecomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 62, 247–262 (1983)
Maejima, M., Sato, K.: 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 Relat. Fields 145, 119–142 (2009)
Maejima, M., Ueda, Y.: Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes. Commun. Stoch. Anal. 3, 349–367 (2009)
Maejima, M., Ueda, Y.: Compositions of mappings of infinitely divisible distributions with applications to finding the limits of some nested subclasses. Electron. Commun. Probab. 15, 227–239 (2010)
Maejima, M., Ueda, Y.: Nested subclasses of the class of \(\alpha \)-selfdecomposable distributions. Tokyo J. Math. 34, 383–406 (2011)
Rocha-Arteaga, A., Sato, K.: Topics in Infinitely Divisible Distributions and Lévy Processes. Aportaciones Matemáticas, Investigación 17, Sociedad Matemática Mexicana (2003)
Rosinski, J.: On series representations of infinitely divisible random vectors. Ann. Probab. 18, 405–430 (1990)
Rosiński, J.: Series representations of Lévy processes from the perspective of point processes. In: Barndorff-Nielsen, O.E. et al. (ed.) Lévy processes. Theory and Applications, pp. 401–415. Birkhäuser, Boston (2001)
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Sato, K.: Two families of improper stochastic integrals with respect to Lévy processes. ALEA Lat. Am. J. Probab. Math. Stat. 1, 47–87 (2006)
Sato, K.: Transformations of infinitely divisible distributions via improper stochastic integrals. ALEA Lat. Am. J. Probab. Math. Stat. 3, 67–110 (2007)
Sato, K.: Fractional integrals and extensions of selfdecomposability. In: Lecture Notes in Mathematics 2001, Lévy matters I, pp. 1–91. Springer, Berlin (2010)
Sato, K.: Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions. ALEA Lat. Am. J. Probab. Math. Stat. 8, 1–17 (2011)
Sato, K.: Inversions of infinitely divisible distributions and conjugates of stochastic integral mappings. J. Theoret, Probab (2013) (To appear)
Sato, K., Ueda, Y.: Weak drifts of infinitely divisible distributions and their applications. J. Theoret. Probab. 26, 885–898 (2013)
Sato, K., Yamazato, M.: Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stoch. Proc. Appl. 17, 73–100 (1984)
Wolfe, S.J.: On a continuous analogue of the stochastic difference equation \(X_n=\rho X_{n-1}+B_n\). Stoch. Proc. Appl. 12, 301–312 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Makoto Maejima’s research was partially supported by JSPS Grant-in-Aid for Science Research 22340021. Jan Rosiński’s research was partially supported by the Simons Foundation grant 281440.
Rights and permissions
About this article
Cite this article
Maejima, M., Rosiński, J. & Ueda, Y. Stochastic Integral and Series Representations for Strictly Stable Distributions. J Theor Probab 28, 989–1006 (2015). https://doi.org/10.1007/s10959-013-0518-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-013-0518-8