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 Xi(θi), i =1, ..., n, be independent random variables such that the distribution of Xi(θi) is determined by the value of θi. Let S(θ) = (X1(θ1), X1(θ1) + X2(θ2), ..., Σni = 1Xi(θi)). We give sufficient conditions on f : Rn → R and on {Xi(θ), θ ∈ Θ} under which f(S(θ)) have some stochastic arrangement monotonicity and stochastic Schur-convexity properties.