Abstract
A collection of random variables {X(θ), θ∈Θ} is said to be parametrically stochastically increasing and convex (concave) in θ∈Θ if X(θ) is stochastically increasing in θ, and if for any increasing convex (concave) function ϕ, Eϕ(X(θ)) is increasing and convex (concave) in θ∈Θ whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X n(θ), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X n(θ), θ∈Θ}, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.
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Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.
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Shaked, M., Shanthikumar, J.G. Parametric stochastic convexity and concavity of stochastic processes. Ann Inst Stat Math 42, 509–531 (1990). https://doi.org/10.1007/BF00049305
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DOI: https://doi.org/10.1007/BF00049305