On Geometric Infinite Divisibility

Abstract. The notion of geometric version of an infinitely divisible law is in-troduced. Concepts parallel to attraction and partial attraction are developedand studied in the setup of geometric summing of random variables. 1. InroductionKlebanov, etal. [5] dened;Definition 1.1. A random variable (r.v) X is geometrically infinitely divisible(GID) if for every p ∈ (0,1), X = d X (p)1 + ... + X p )N p , where N p and {X (pi } areindependent, {X (p)i } are i.i.d and N p is a geometric r.v with mean 1/p.Equivalently, a r.v X with characteristic function (c.f) φ(t) is GID if and only ifexp{1−1/φ(t)} is an infinitely divisible (ID) c.f. They also introduced geometricallystrictly stable (GSS) laws as:Definition 1.2. A r.v Y is GSS if for every p ∈ (0,1), there exists a constantc(p) > 0 such that Y = d c(p){X 1 +...+X N p }, where N p and {X i } are independent,{X i } are i.i.d, Y= d X 1 and N p is a geometric r.v with mean 1/p.Pillai [9] introduced semi-α-Laplace laws as:Definition 1.3. A distribution function (d.f) F with c.f φ(t) = 1/(1 + g(t)) issemi-α-Laplace of exponent α, 0 < α ≤ 2, if g(t) = ag(bt) for some constants a andb, 0 < b < 1 < a and α is the unique solution of ab