Parametric stochastic convexity and concavity of stochastic processes |
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Authors: | Moshe Shaked J George Shanthikumar |
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Institution: | (1) Department of Mathematics, University of Arizona, 85721 Tucson, AZ, U.S.A.;(2) School of Business Administration, University of California, 94720 Berkeley, CA, U.S.A. |
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Abstract: | A collection of random variables {X( ), ![theta](/content/r684617337044722/xxlarge952.gif) ![isin](/content/r684617337044722/xxlarge8712.gif) } is said to be parametrically stochastically increasing and convex (concave) in ![theta](/content/r684617337044722/xxlarge952.gif) ![isin](/content/r684617337044722/xxlarge8712.gif) if X( ) is stochastically increasing in , and if for any increasing convex (concave) function , E (X( )) is increasing and convex (concave) in ![theta](/content/r684617337044722/xxlarge952.gif) ![isin](/content/r684617337044722/xxlarge8712.gif) 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( ), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X
n( ), ![theta](/content/r684617337044722/xxlarge952.gif) ![isin](/content/r684617337044722/xxlarge8712.gif) }, 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.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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Keywords: | Sample path convexity and concavity Markov processes directional convexity and concavity single stage queues supermodular and submodular functions L-superadditive functions reliability theory branching processes shock models total positivity |
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