## Electronic Journal of Probability

### A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class

#### Abstract

Let $\{X_t^\mu,t\geq0\}$ be a Lévy process on $\mathbb{R}^d$ whose distribution at time $1$ is a $d$-dimensional infinitely distribution $\mu$. It is known that the set of all infinitely divisible distributions on $\mathbb{R}^d$, each of which is represented by the law of a stochastic integral $\int_0^1\!\log(1/t)\,dX_t^\mu$ for some infinitely divisible distribution on $\mathbb{R}^d$, coincides with the Goldie-Steutel-Bondesson class, which, in one dimension, is the smallest class that contains all mixtures of exponential distributions and is closed under convolution and weak convergence. The purpose of this paper is to study the class of infinitely divisible distributions which are represented as the law of $\int_0^1\!(\log(1/t))^{1/\alpha}\,dX_t^\mu$ for general $\alpha>0$. These stochastic integrals define a new family of mappings of infinitely divisible distributions. We first study properties of these mappings and their ranges. Then we characterize some subclasses of the range by stochastic integrals with respect to some compound Poisson processes. Finally, we investigate the limit of the ranges of the iterated mappings.

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 35, 1119-1142.

Dates
Accepted: 7 July 2010
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819820

Digital Object Identifier
doi:10.1214/EJP.v15-791

Mathematical Reviews number (MathSciNet)
MR2659759

Zentralblatt MATH identifier
1225.60026

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions

Rights

#### Citation

Aoyama, Takahiro; Lindner, Alexander; Maejima, Makoto. A New Family of Mappings of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class. Electron. J. Probab. 15 (2010), paper no. 35, 1119--1142. doi:10.1214/EJP.v15-791. https://projecteuclid.org/euclid.ejp/1464819820

#### References

• T. Aoyama. Nested subclasses of the class of type G selfdecomposable distributions on Rd. Probab. Math. Statist. 29 (2009), 135-154.
• T. Aoyama, M. Maejima, and J. RosiÅ„ski. A subclass of type G selfdecomposable distributions on Rd. J. Theor. Probab. 21 (2008), 14-34.
• 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.
• L. Bondesson. Class of infinitely divisible distributions and densities. Z. Wahrsch. Verw. Gebiete 57 (1981),39-71; Correction and addendum 59 (1982), 277.
• W. Feller. An introduction to probability theory and its applications, Vol. II, 2nd ed. John Wiley & Sons (1981).
• L.F. James, B. Roynette and M. Yor. Generalized Gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5 (2008), 346-415.
• M. Maejima. Subclasses of Goldie-Steutel-Bondesson class of infinitely divisible distributions on Rd by Î¥-mapping. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 55-66.
• M. Maejima and G. Nakahara. A note on new classes of infinitely divisible distributions on Rd. Electron. Commun. Probab. 14 (2009), 358-371.
• 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 Relat. Fields 145 (2009), 119-142.
• K. Sato. Class L of multivariate distributions and its subclasses. J. Multivariate Anal. 10 (1980), 207-232.
• K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999).
• K. Sato. Stochastic integrals in additive processes and application to semi-Lévy processes. Osaka J. Math. 41 (2004), 211-236.
• K. Sato. Additive processes and stochastic integrals. Illinois J. Math. 50 (2006). 825-851.
• K. Sato. Two families of improper stochastic integrals with respect to Lévy processes. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 47-87.
• K. Sato. Transformations of infinitely divisible distributions via improper stochastic integrals. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007), 67-110.
• S.J. Wolfe. On a continuous analogue of the stochastic difference equation Xn=Ï Xn-1+Bn. Stochastic Process. Appl. 12 (1982), 301-312.