ON GEOMETRIC-STABLE LAWS, A RELATED PROPERTY OF STABLE PROCESSES, AND STABLE DENSITIES

OF EXPONENT ONE

B. RAMACHANDRAN

Indian Statistical Institute, Delhi Centre, New Delhi-16, India

(Received November 8, 1995; revised May 13, 1996)

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).

Key words and phrases: Stable laws and processes, geometric-stable laws, geometric domains of attraction.