Polya Type Distributions of Convolutions

Introduction. The theory of totally positive kernels and P6lya type distributions has been decisively and extensively applied in several domains of mathematics, statistics, economics and mechanics. Totally positive kernels arise naturally in developing procedures for inverting, by differential polynomial operators [7], integral transformations defined in terms of convolution kernels. The theory of P6lya type distributions is fundamental in permitting characterizations of best statistical procedures for decision problems [8] [9] [13]. In clarifying the structure of stochastic processes with continuous path functions we encounter totally positive kernels [111 [12]. Studies in the stability of certain models in mathematical economics frequently use properties of totally positive kernels [10]. The theory of vibrations of certain types of mechanical systems (primarily coupled systems) involves aspects of the theory of totally positive kernels [5]. In this paper, we characterize new classes of totally positive kernels that arise from summing independent random variables and forming related first passage time distributions. A function f(x, y) of two real variables ranging over linearly ordered one dimensional sets X and Y respectively, is said to be totally positive of order k (TPk) if for allx < X2 < ... < Xm.yy < y2< ... < ym, (xieX;yjEY) and all 1 < m < k,