# A NOTE ON SEMI-SELFDECOMPOSABILITY AND OPERATOR SEMI-STABILITY IN SUBORDINATION

• Choi, Gyeong-Suk (Department of mathematics Kangwon National University) ;
• Kim, Yun-Kyong (Department of Information & Communication Engineering Dongshin University) ;
• Joo, Sang-Yeol (Department of mathematics Kangwon National University)
• Published : 2010.05.31

#### Abstract

Some results on inheritance of operator semi-selfdecomposability and its decreasing subclass property from subordinator to subordinated in subordination of a L$\acute{e}$evy process are given. A main result is an extension of results of [5] to semi-selfdecomposable subordinator. Its consequence is discussed.

#### References

1. O. E. Barndorff-Nielsen, M. Maejima, and K. Sato, Infinite divisibility for stochastic processes and time change, J. Theoret. Probab. 19 (2006), no. 2, 411-446. https://doi.org/10.1007/s10959-006-0020-7
2. O. E. Barndorff-Nielsen, J. Pedersen, and K. Sato, Multivariate subordination, self-decomposability and stability, Adv. in Appl. Probab. 33 (2001), no. 1, 160-187. https://doi.org/10.1239/aap/999187902
3. G. S. Choi, Criteria for recurrence and transience of semistable processes, Nagoya Math. J. 134 (1994), 91-106.
4. G. S. Choi, Characterization of strictly operator semi-stable distributions, J. Korean Math. Soc. 38 (2001), no. 1, 101-123.
5. G. S. Choi, Some results on subordination, selfdecomposability and operator semi-stability, Statist. Probab. Lett. 78 (2008), no. 6, 780-784. https://doi.org/10.1016/j.spl.2007.09.044
6. G. S. Choi, S. Y. Joo, and Y. K. Kim, Subordination, self-decomposability and semistability, Commun. Korean Math. Soc. 21 (2006), no. 4, 787-794. https://doi.org/10.4134/CKMS.2006.21.4.787
7. C. Halgreen, Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions, Z. Wahrsch. Verw. Gebiete 47 (1979), no. 1, 13-17. https://doi.org/10.1007/BF00533246
8. R. Jajte, Semi-stable probability measures on RN, Studia Math. 61 (1977), no. 1, 29-39.
9. T. J. Kozubowski, A note on self-decomposability of stable process subordinated to self-decomposable subordinator, Statist. Probab. Lett. 73 (2005), no. 4, 343-345 https://doi.org/10.1016/j.spl.2005.04.011
10. T. J. Kozubowski, A note on self-decomposability of stable process subordinated to self-decomposable subordinator, Statist. Probab. Lett. 74 (2005), no. 1, 89-91. https://doi.org/10.1016/j.spl.2005.04.025
11. A. Luczak, Operator semistable probability measures on $R^{N}$, Colloq. Math. 45 (1981), no. 2, 287-300
12. A. Luczak, Operator semistable probability measures on $R^{N}$, Corrigenda, Colloq. Math. 52 (1987), no. 1, 167-169.
13. M. Maejima and Y. Naito, Semi-selfdecomposable distributions and a new class of limit theorems, Probab. Theory Related Fields 112 (1998), no. 1, 13-31. https://doi.org/10.1007/s004400050181
14. M. Maejima, K. Sato, and T. Watanabe, Operator semi-selfdecomposability, (C,Q)-decomposability and related nested classes, Tokyo J. Math. 22 (1999), no. 2, 473-509. https://doi.org/10.3836/tjm/1270041450
15. M. Maejima, K. Sato, and T. Watanabe, Completely operator semi-selfdecomposable distributions, Tokyo J. Math. 23 (2000), no. 1, 235-253. https://doi.org/10.3836/tjm/1255958818
16. M. Maejima, K. Sato, and T. Watanabe, Distributions of selfsimilar and semi-selfsimilar processes with independent increments, Statist. Probab. Lett. 47 (2000), no. 4, 395-401. https://doi.org/10.1016/S0167-7152(99)00184-4
17. J. Pedersen and K. Sato, Cone-parameter convolution semigroups and their subordination, Tokyo J. Math. 26 (2003), no. 2, 503-525. https://doi.org/10.3836/tjm/1244208605
18. J. Pedersen and K. Sato, Relations between cone-parameter Levy processes and convolution semigroups, J. Math. Soc. Japan 56 (2004), no. 2, 541-559. https://doi.org/10.2969/jmsj/1191418644
19. K. Sato, Levy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.
20. K. Sato, Subordination and self-decomposability, Statist. Probab. Lett. 54 (2001), no. 3, 317-324. https://doi.org/10.1016/S0167-7152(01)00110-9
21. K. Sato, Selfdecomposability and semi-selfdecomposability in subordination of coneparameter convolution semigroups, Tokyo J. Math. 32 (2009), no. 1, 81-90. https://doi.org/10.3836/tjm/1249648410