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June, 1976 Note on the $k$-Dimensional Jensen Inequality
Martin Schaefer
Ann. Probab. 4(3): 502-504 (June, 1976). DOI: 10.1214/aop/1176996102


Let $f$ be a measurable convex function from $R^k$ to $R^1$ and let $X_1, \cdots, X_k$ be real-valued integrable random variables. The best approximation for $f(EX_1, \cdots, EX_k)$ one can get by Jensen's inequality is $f(EX_1, \cdots, EX_k) \leqq \inf Ef(\mathbf{Z})$ where the infimum is taken over all $k-\dim$. random vectors $\mathbf{Z} = (Z_1, \cdots, Z_k)'$ such that $Z_i$ has the same distribution as $X_i (1 \leqq i \leqq k)$. An application is given in the case where $f(y)$ is the span of the vector $y$ which leads to a new approximation for $f(A\mathbf{u})$ where $A$ is a stochastic $(k \times m)$-matrix and $\mathbf{u}$ is an arbitrary element of $R^m$.


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Martin Schaefer. "Note on the $k$-Dimensional Jensen Inequality." Ann. Probab. 4 (3) 502 - 504, June, 1976.


Published: June, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0342.26019
MathSciNet: MR400339
Digital Object Identifier: 10.1214/aop/1176996102

Primary: 52A40

Keywords: convex function , Jensen inequality

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 3 • June, 1976
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