## Tokyo Journal of Mathematics

### Selfdecomposability and Semi-selfdecomposability in Subordination of Cone-parameter Convolution Semigroups

Ken-iti SATO

#### Abstract

Extension of two known facts concerning subordination is made. The first fact is that, in subordination of $1$-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\mathbf{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.

#### Article information

Source
Tokyo J. Math., Volume 32, Number 1 (2009), 81-90.

Dates
First available in Project Euclid: 7 August 2009

https://projecteuclid.org/euclid.tjm/1249648410

Digital Object Identifier
doi:10.3836/tjm/1249648410

Mathematical Reviews number (MathSciNet)
MR2541155

Zentralblatt MATH identifier
1180.60065

#### Citation

SATO, Ken-iti. Selfdecomposability and Semi-selfdecomposability in Subordination of Cone-parameter Convolution Semigroups. Tokyo J. Math. 32 (2009), no. 1, 81--90. doi:10.3836/tjm/1249648410. https://projecteuclid.org/euclid.tjm/1249648410

#### References

• O. E. Barndorff-Nielsen, J. Pedersen and K. Sato, Multivariate subordination, selfdecomposability and stability, Adv. Appl. Probab., 33 (2001), 160–187.
• O. E. Barndorff-Nielsen and V. Pérez-Abreu, Extensions of type $G$ and marginal infinite divisibility, Theory Probab. Appl., 47 (2003), 202–218.
• B. Grigelionis, On subordinated multivariate Gaussian Lévy processes, Acta Appl. Math., 96 (2007), 233–246.
• C. Halgreen, Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions, Zeit. Wahrsch. Verw. Gebiete, 47 (1979), 13–17.
• H. Kondo, M. Maejima and K. Sato, Some properties of exponential integrals of Lévy processes and examples, Elect. Comm. in Probab., 11 (2006), 291–303.
• T. J. Kozubowski, A note on self-decomposability of stable process subordinated to self-decomposable subordinator, Stat. Probab. Let., 73 (2005), 343–345 and 74 (2005), 89–91.
• A. Lindner and K. Sato, Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein-Uhlenbeck processes, Ann. Probab., 37 (2009), 250–274.
• M. Maejima and Y. Naito, Semi-selfdecomposable distributions and a new class of limit theorems, Probab. Theory Relat. Fields, 112 (1998), 13–31.
• M. Maejima and K. Sato, Semi-Lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein-Uhlenbeck type processes, J. Math. Kyoto Univ., 43 (2003), 609–639.
• M. Maejima, K. Sato and T. Watanabe, Operator semi-selfdecomposability, $(C,Q)$-decomposability and related nested classes, Tokyo J. Math., 22 (1999), 473–509.
• J. Pedersen and K. Sato, Cone-parameter convolution semigroups and their subordination, Tokyo J. Math., 26 (2003), 503–525.
• J. Pedersen and K. Sato, Relations between cone-parameter Lévy processes and convolution semigroups, J. Math. Soc. Japan, 56 (2004), 541–559.
• B. Ramachandran, On geometric-stable laws, a related property of stable processes, and stable densities of exponent one, Ann. Inst. Statist. Math., 49 (1997), 299–313.
• K. Sato, Multivariate distributions with selfdecomposable projections, J. Korean Math. Soc., 35 (1998), 783–791.
• K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, 1999.
• K. Sato, Subordination and self-decomposability, Stat. Probab. Let., 54 (2001), 317–324.
• A. V. Skorohod, Random Processes with Independent Increments, Kluwer Academic Pub., 1991.
• K. Takano, On mixtures of the normal distribution by the generalized gamma convolutions, Bull. Fac. Sci. Ibaraki Univ. Ser. A, 21 (1989), 29–41; Correction and addendum, 22 (1990), 49–52.
• T. Watanabe, Absolute continuity of some semi-selfdecomposable distributions and self-similar measures, Probab. Theory Relat. Fields, 117 (2000), 387–405.