Hostname: page-component-f554764f5-sl7kg Total loading time: 0 Render date: 2025-04-09T02:40:33.398Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillation of modes of some semi-stable Lévy processes

Published online by Cambridge University Press:  22 January 2016

Toshiro Watanabe
Toshiro Watanabe*
Affiliation:
Center for Mathematical Sciences, The University of Aizu, IkkimachiAizuwakamatsu, Fukushima, 965, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 132 , December 1993 , pp. 141 - 153
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[1] Kagan, A. M., Linnik, Yu. V. and Rao, C. R., Characterization Problems in Mathematical Statistics, John Wiley & Sons, 1973.Google Scholar
[2] Lévy, P., Théorie de l’addition des variables aléatoires, ème ed. (lère éd. 1937), Gauthier-Villars, Paris, 1954.Google Scholar
[3] Medgyessy, P., On a new class of unimodal infinitely divisible distribution functions and related topics, Studia Sci. Math. Hangar., 2 (1967), 441–446.Google Scholar
[4] Sato, K., Bounds of modes and unimodal processes with independent increments, Nagoya Math. J., 104 (1986), 29–42.Google Scholar
[5] Sato, K., Behavior of modes of a class of processes with indecpendent increments, J. Math. Soc. Japan, 38 (1986), 679–695.Google Scholar
[6] Sato, K., On unimodality and mode behavior of Levy processes, “Probability Theory and Mathematical Statistics. Proceedings of the Sixth USSR-Japan Symposium”edited by Shiryaev, A. N. et al., World Scientific, Singapore, 1992, pp.292–305.Google Scholar
[7] Sato, K. and Yamazato, M., On distribution functions of class L, Z. Wahrsch. verw. Gebiete, 43 (1978), 273–308.Google Scholar
[8] Steutel, F. W. and van Harn, K., Discrete analogues of self-decomposability and stability, Ann. Probability, 7 (1979), 893–899.Google Scholar
[9] Watanabe, T., Non-symmetric unimodal Levy processes that are not of class L , Japan. J. Math., 15 (1989), 191–203.CrossRefGoogle Scholar
[10] Watanabe, T., On the strong unimodality of Levy processes, Nagoya Math. J., 121 (1991), 195–199.Google Scholar
[11] Watanabe, T., On unimodal Lévy processes on the nonnegative integers, J. Math. Soc. Japan, 44 (1992), 239–250.Google Scholar
[12] Watanabe, T., On Yamazato’s property of unimodal one-sided Levy processes, Kodai Math. J., 15 (1992), 50–64.Google Scholar
[13] Watanabe, T., Sufficient conditions for unimodality of non-symmetric Levy processes, Kodai Math. J., 15 (1992), 82–101.Google Scholar
[14] Wolfe, S. J., On the unimodality of L functions, Ann. Math. Statist, 42 (1971), 912–918.Google Scholar
[15] Wolfe, S. J., On the unimodality of infinitely divisible distribution functions, Z. Wahrsch. verw. Gebiete, 45 (1978), 329–335.Google Scholar
[16] Yamazato, M., Unimodality of infinitely divisible distribution functions of class L, Ann. Probability, 6 (1978), 523–531.CrossRefGoogle Scholar
[17] Yamazato, M., On strongly unimodal infinitely divisible distributions, Ann. Probability, 10 (1982), 589–601.Google Scholar
[18] Yamazato, M., Characterization of the class of upward first passage time distributions of birth and death processes and related results, J. Math. Soc. Japan, 40 (1988), 477–499.Google Scholar
[19] Yamazato, M., On subclasses of infinitely divisible distributions on R related to hitting time distributions of 1-dimensional generalized diffusion processes, Nagoya Math. J., 127 (1992), 175–200.Google Scholar
[20] Yamazato, M., On strongly unimodal infinitely divisible distributions of class CME, Pre print.Google Scholar