It is well known (see [2], p. 158) that if X and Y are independent random variables with a continuous joint probability density function (pdf) which is spherically symmetric about the origin, then both X and Y are normally distributed. In this note we examine the condition that the joint pdf be spherically symmetric about the origin and show that the normal distribution is strongly dependent on the choice of metric for R2.

The paper uses Lévy processes and bivariate Lévy copulae in order to model the behavior of intraday log-returns. Based on assumptions about the form of marginal tail integrals and a Clayton Lévy copula, the model allows for capturing intraday cross-dependency. The model is applied to VaR of the portfolios constructed on stock returns as well as on cryptocurrencies. The proposed method shows fair performance compared to classical time series models.

Observations are made on a point process $𝛯$ in ${ℝ}^d$ in a window $Q_{\lambda }$ of volume $\lambda$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,𝛯)$, is a function of the points within a random distance of $x$. When the input $𝛯$ is a Poisson or binomial point process, the large $\lambda$ limit theory for the total score ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input $𝛯$....

In the framework of models generated by compositional expressions, we solve two topical marginalization problems (namely, the single-marginal problem and the marginal-representation problem) that were solved only for the special class of the so-called “canonical expressions”. We also show that the two problems can be solved “from scratch” with preliminary symbolic computation.

Efficient computational algorithms are what made graphical Markov models so popular and successful. Similar algorithms can also be developed for computation with compositional models, which form an alternative to graphical Markov models. In this paper we present a theoretical basis as well as a scheme of an algorithm enabling computation of marginals for multidimensional distributions represented in the form of compositional models.

We study the problem of finding the smallest $m$ such that every element of an exponential family can be written as a mixture of $m$ elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that $m=q^{N-1}$ is the smallest number for which any distribution of $N$$q...$

We calculate the moments $m_{k,0}$ of the measure orthogonalizing the 2-dimensional Chebyshev polynomials introduced by Koornwinder.

In the paper the authors investigate the explicit form of the joint Laplace transform of the distances between two subsequent moments f particle registrations by the Type II counter (the counter with prolonged dead time), in the general case, and the generating function of the number of particles arriving during the dead time. They give explicit solutions to the complicated integral equations obtained by L. Takács and R. Pyke, respectively. Moreover, they study the geometric behaviour of the distribution...

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.