Note on the $k$-Dimensional Jensen Inequality
1976; Institute of Mathematical Statistics; Volume: 4; Issue: 3 Linguagem: Inglês
10.1214/aop/1176996102
ISSN2168-894X
Autores Tópico(s)Functional Equations Stability Results
ResumoLet $f$ 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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