Abstract
Klebanov et al. (1985, Theory Probab. Appl., 29, 791-794) introduced a class of probability laws termed by them "geometrically-infinitely-divisible" laws, and studied in detail the sub-class of "geometrically-strictly-stable" laws. In Section 2 of the present paper, the larger sub-class of "geometric-stable" laws is (defined and) studied. In Section 3, a characterization of stable processes involving (stochastic integrals and) geometric-stable laws is presented. In Section 4, the asymptotic behaviour of stable densities of exponent one (and |β| < 1) is studied using only real analysis methods. In an Appendix, "geometric domains of attraction" to geometric-stable laws are investigated, motivated by the work of Mohan et al. (1993, Sankhyā Ser. A, 55, 171-179).
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Ramachandran, B. On Geometric-Stable Laws, a Related Property of Stable Processes, and Stable Densities of Exponent One. Annals of the Institute of Statistical Mathematics 49, 299–313 (1997). https://doi.org/10.1023/A:1003115013644
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DOI: https://doi.org/10.1023/A:1003115013644