Another Look at Random Infinite Divisibility

The drawbacks in the formulations of random infinite divisibility in Sandhya (1991, 1996), Gnedenko and Korelev (1996), Klebanov and Rachev (1996), Bunge (1996) and Kozubowski and Panorska (1996) are pointed out. For any given Laplace transform, we conceive random (N) infinite divisibility w.r.t a class of probability generating functions derived from the Laplace transform itself. This formulation overcomes the said drawbacks, and the class of probability generating functions is useful in transfer theorems for sums and maximums in general. Generalizing the concepts of attraction (and partial attraction) in the classical and the geometric summation setup to our formulation we show that the domains of attraction (and partial attraction)in all these setups are same. We also establish a necessary and sufficient condition for the convergence to infinitely divisible laws from that of an N-sum and conversely, that is an analogue of Theorem.4.6.5 in Gnedenko and Korelev (1996, p.149). The role of the divisibiltiy of N and the Laplace transform on that of this formulation is also discussed.