## The Annals of Probability

### Note on the $k$-Dimensional Jensen Inequality

Martin Schaefer

#### Abstract

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 $R^m$.

#### Article information

Source
Ann. Probab., Volume 4, Number 3 (1976), 502-504.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996102

Digital Object Identifier
doi:10.1214/aop/1176996102

Mathematical Reviews number (MathSciNet)
MR400339

Zentralblatt MATH identifier
0342.26019

JSTOR
Schaefer, Martin. Note on the $k$-Dimensional Jensen Inequality. Ann. Probab. 4 (1976), no. 3, 502--504. doi:10.1214/aop/1176996102. https://projecteuclid.org/euclid.aop/1176996102