In this paper we derive conditions upon the nonnegative random variable under which the inequality $Dg\left(\xi \right)\le cE{\left[g^{\text{'}}\left(\xi \right)\right]}^2\xi$ holds for a fixed nonnegative constant $c$ and for any absolutely continuous function $g$. Taking into account the characterization of a Gamma distribution we consider the functional $U_{\xi }={sup}_g\frac{Dg\left(\xi \right)}{E{\left[g^{\text{'}}\left(\xi \right)\right]}^2\xi }$ and establishing some of its properties we show that $U_{\xi }\ge 1$ and that $U_{\xi }=1$ iff the random variable $\xi$ has a Gamma distribution.

In this paper two characterizations of the Pólya distribution are obtained when its contagion parameter is negative. One of them is based on mixtures and the other one is obtained by characterizing a subfamily of the discrete Pearson system.

In this paper two characterizations of the Pólya distribution are obtained when its contagion parameter is negative. One of them is based on mixtures and the other one is obtained by characterizing a subfamily of the discrete Pearson system.

We consider representations of a joint distribution law of a family of categorical random variables (i.e., a multivariate categorical variable) as a mixture of independent distribution laws (i.e. distribution laws according to which random variables are mutually independent). For infinite families of random variables, we describe a class of mixtures with identifiable mixing measure. This class is interesting from a practical point of view as well, as its structure clarifies principles of selecting...

We introduce the concept of monotone dependence function of bivariate distributions without moment conditions. Our concept gives, among other things, a characterization of independent and positively (negatively) quadrant dependent random variables.

Let X be a countable discrete Abelian group, Aut(X) the set of automorphisms of X, and I(X) the set of idempotent distributions on X. Assume that α₁, α₂, β₁, β₂ ∈ Aut(X) satisfy $\beta ₁\alpha {₁}^{-1}\pm \beta ₂\alpha {₂}^{-1}\in Aut\left(X\right)$. Let ξ₁, ξ₂ be independent random variables with values in X and distributions μ₁, μ₂. We prove that the symmetry of the conditional distribution of L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ implies that μ₁, μ₂ ∈ I(X) if and only if the group X contains no elements of order two. This theorem can be considered as an analogue...

We establish the optimal quantization problem for probabilities under constrained Rényi-$\alpha$-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases $\alpha =0$ (restricted codebook size) and $\alpha =1$ (restricted Shannon entropy).