Karlin's Basic Composition Theorems and Stochastic Orderings

Suppose λ, x, ζ traverse the ordered sets Λ, X and Z, respectively and consider the functions f(λ, x, ζ) and g(λ, ζ) satisfying the following conditions, (a) f(λ, x, ζ) > 0 and f is TP2 in each pairs of variables when the third variable is held fixed; and (b) g(λ, ζ) is TP2. Then the function h(λ, x) =∫Zf(λ, x, ζ)g(λ, ζ)dμ(ζ), defined on Λ X is TP2 in (λ, x). The aim of this note is to use a new stochastic ordering argument to prove the above result and simplify it’s proof given by Karlin (1968). We also prove some other new versions of this result