### A Characteristic Property for Maxima of Random Variables with Trivial Analogue When Sums are Concerned

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In this paper we propose a new generalized Rayleigh distribution different from that introduced in Apl. Mat. 47 (1976), pp. 395–412. The construction makes use of the so-called “conservability approach” (see Kybernetika 25 (1989), pp. 209–215) namely, if $X$ is a positive continuous random variable with a finite mean-value $E\left(X\right)$, then a new density is set to be ${f}_{1}\left(x\right)=xf\left(x\right)/E\left(X\right)$, where $f\left(x\right)$ is the probability density function of $X$. The new generalized Rayleigh variable is obtained using a generalized form of the exponential...

A characterization of geometric distribution is given, which is based on the ratio of the real and imaginary part of the characteristic function.

The class of extended Pólya functions Ω = {φ: φ is a continuous real valued real function, φ(-t) = φ(t) ≤ φ(0) ∈ [0,1], límt→∞ φ(t) = c ∈ [0,1] and φ(|t|) is convex} is a convex set. Its extreme points are identified, and using Choquet's theorem it is shown that φ ∈ Ω has an integral representation of the form φ(|t|) = ∫0∞ max{0, 1-|t|y} dG(y), where G is the distribution function of some random variable Y. As on the other hand max{0, 1-|t|y} is the characteristic function of an absolutely continuous...

We derive a necessary condition for stochastic dominance of any order based on the Laplace transform of probability measures on [0,∞) for which it follows easily Fishburn's theorem on the lexicographic order of the moments.

In this paper, we consider a symmetric α-stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function $exp-\left(\right|a|p+{\left|b\right|p)}^{\alpha /p}$ is a characteristic function of such a vector for some p and α. The solution of this problem we can find in [3], in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in [4].

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1-\Phi \left(z\right)$ inside the unit disk, where $\Phi $ is the generating function of $X$, under some mild conditions