Characterization of probability distributions by Poincaré-type inequalities
The author considers generalizations of the following characterization of the normal distribution, due to \textit{A. A. Borovkov} and \textit{S. A. Utev} [Teor. Veroyatn. Primen. 28, No.2, 209-218 (1983; Zbl 0511.60016); English translation in Theory Probab. Appl. 28, 219-228 (1984)]: \[ (*)\quad U_ X:=\sup_{g}\frac{var g(X)}{\sigma^ 2E(g'(X))^ 2}=1 \] if and only if X has a normal distribution. When X is taken to be m- dimensional and the denominator of \(U_ X\) is generalized such as to include a positive definite \(m\times m\) matrix \(\Sigma\) and a measure \(\mu\), then the analogue of (*) characterizes the infinitely divisible distributions having a normal component with covariance matrix \(\Sigma\) and canonical measure \(\mu\) ; this is the content of Theorem 2.1. Related results include a limit version of Theorem 2.1, more general characterization theorems, and some discrete analogues.
characterization of the normal distribution
scientific article; zbMATH DE number 3990490