A convexity property of expectations under exponential weights

Marton Balazs, Timo Seppalainen

Research output: Contribution to journalArticle (Academic Journal)

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Abstract

Take a random variable X with some finite exponential moments. Define an exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for admissible values of the parameter t. Denote the weighted expectation of X itself by r(t) = E^t(X), with inverse function t(r). We prove that for a convex function f the expectation E^{t(r)}(f) is a convex function of the parameter r. Along the way we develop correlation inequalities for convex functions. Motivation for this result comes from equilibrium investigations of some stochastic interacting systems with stationary product distributions. In particular, convexity of the hydrodynamic flux function follows in some cases.
Original languageEnglish
Number of pages12
JournalarXiv
Publication statusPublished - 30 Jul 2007

Bibliographical note

After completion of this manuscript we learned that our main results can be obtained as a special case of some propositions in Karlin: Total Positivity, Vol.1

Keywords

  • math.PR
  • 60K35, 60E15

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