The generalized Lorenz order and the absolute Lorenz order are used in economics to compare income distributions in terms of social welfare. In Section 2, we show that these orders are equivalent to two stochastic orders, the concave order and the dilation order, which are used to compare the dispersion of probability distributions. In Section 3, a sufficient condition for the absolute Lorenz order, which is often easy to verify in practice, is presented. This condition is applied in Section 4 to...

Two characterizations of the exponential distribution among distributions with support the nonnegative real axis are presented. The characterizations are based on certain properties of the characteristic function of the exponential random variable. Counterexamples concerning more general possible versions of the characterizations are given.

In this paper we give a characterization of the multivariate normal distribution through the conditional distributions in the most general case, which include the singular distribution.

Let (X1,Y1),...,(Xm,Ym) be m independent identically distributed bivariate vectors and L1 = β1X1 + ... + βmXm, L2 = β1X1 + ... + βmXm are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of L1 and L2 imply the same property for X1 and Y1, and under what conditions does the independence of L1 and L2 entail independence of X1 and Y1? Some analytical sufficient conditions are obtained and it is shown that in general they can not be...

Let $(X_1,Y_1),...,(X_m,Y_m)$ be $m$ independent identically distributed bivariate vectors and $L_1={\beta }_1{X}_1+...+{\beta }_m{X}_m$, $L_2={\beta }_1{Y}_1+...+{\beta }_m{Y}_m$ are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of $L_1$ and $L_2$ imply the same property for $X_1$ and $Y_1$, and under what conditions does the independence of $L_1$ and $L_2$ entail independence of $X_1$ and $Y_1$? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.

Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.

A toute mesure $c$ positive sur ${ℝ}_{+}$ telle que ${\int }_0^{\infty }(x\wedge x^2)c\left(dx\right)\<\infty$, nous associons un couple de Wald indéfiniment divisible, i.e. un couple de variables aléatoires $(X,H)$ tel que $X$ et $H$ sont indéfiniment divisibles, $H\ge 0$, et pour tout $\lambda \ge 0,E\left(e^{\lambda X}\right)·E\left(e^{-\frac{{\lambda }^2}{2}H}\right)=1$. Plus généralement, à une mesure $c$ positive sur ${ℝ}_{+}$ telle que ${\int }_0^{\infty }{e}^{-\alpha x}{x}^{2}c\left(dx\right)\<\infty$ pour tout $\alpha \>{\alpha }_0$, nous associons une “famille d’Esscher” de couples de Wald indéfiniment divisibles. Nous donnons de nombreux exemples de telles familles d’Esscher. Celles liées à la fonction gamma et à la fonction zeta de Riemann possèdent...