PARAMETRIC SCHUR CONVEXITY AND ARRANGEMENT MONOTONICITY PROPERTIES OF PARTIAL-SUMS

Studying the joint distributional properties of partial sums of independent random variables, we obtain stochastic analogues of some simple deterministic results from the theory of majorization, Schur-convexity, and arrangement monotonicity. More explicitly, let X(i)(theta(i)), i=1, ..., n, be independent random variables such that the distribution of X(i)(theta(i)) is determined by the value of theta(i). Let S(theta) = (X(1)(theta(1)), X(1)(theta(1)) + X(2)(theta(2)), ..., Sigma(i=1)(n) X(i)(theta(i))). We give sufficient conditions on f:R(n) --> R and on {X(i)(theta), theta is an element of Theta} under which f(S(theta)) have some stochastic arrangement monotonicity and stochastic Schur-convexity properties. (C) 1995 Academic Press, Inc.