Abstract
We investigate the structure of infinitely divisible probability measures on a discrete linear group. It is shown that for any such measure there is an infinitely divisible elementz in the centralizer of the support of the measure, such that the translate of the measure byz is embeddable over the subgroup generated by the support of the measure. Examples are given to show that this reult is best possible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beardon, A. F. (1983). The Geometry of Discrete Groups, Berlin/Heidelberg/New York, Springer.
Dani, S. G., and McCrudden, M. (1988). Factor toots and embeddability of measures on Lie groups,Math. Z. 199, 369–385.
Dani, S. G., and McCrudden, M. (1988). On the factor set of measures and local rightness of convolution semigroups over Lie groups.J. Th. Prob. 1, 357–370.
Dani, S. G., and McCrudden, M. (1992). Embeddability of infinitely divisible distributions on linear Lie groups.Invent. Math. 110, 237–261.
Dani, S. G., and Shah, Riddhi (1993). On infinitely divisible measures on certain finitely generated groups,Math. Z. 212, 631–636.
Heyer, H. (1977). Probability Measures on Locally Compact Groups, Berlin/Heidelberg/New York, Springer.
Maskit, B. (1971). On Poincaré's theorem for fundamental polygons,Adv. Math. 7, 219–230.
McCrudden, M. (1981). Local tightness of convolution semigroups over locally compact groups, Probability Measures on Groups, Proc. Oberwolfach 1981, Heyer, H. (ed.).Lecture Notes in Mathematics, Vol. 728, Springer-Verlag, Berlin.
Parthasarathy, K. R. (1967).Probability Measures on Metric Spaces, Academic Press, New York.
Selberg, A. (1960). On discontinuous groups in higher-dimensional symmetric spaces, In: Contributions to Function Theory (International Colloquium on Function Theory; Bombay, 1960), pp. 147–164, Tata Inst. of Fundamental Research, Bombay, 1960.
Shah, Riddhi (1991). Infinitely divisible measures onp-adic groups,J. Theor. Prob. 4, 391–405.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dani, S.G., McCrudden, M. Infinitely divisible probabilities on discrete linear groups. J Theor Probab 9, 215–229 (1996). https://doi.org/10.1007/BF02213741
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02213741