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ON GEOMETRIC-STABLE LAWS, A RELATED PROPERTY

OF STABLE PROCESSES, AND STABLE DENSITIES

OF EXPONENT ONE

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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 |*beta*| < 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, *Sankhyã Ser. A*, **55**, 171-179).

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

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