Modern Stochastics: Theory and Applications

A new, direct proof of the formulas for the conic intrinsic volumes of the Weyl chambers of types ${A_{n-1}}$, ${B_{n}}$ and ${D_{n}}$ is given. These formulas express the conic intrinsic volumes in terms of the Stirling numbers of the first kind and their B- and D-analogues. The proof involves an explicit determination of the internal and external angles of the faces of the Weyl chambers.

Let $\{L(t),t\ge 0\}$ be a Lévy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic Lévy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process $Z(t)$ is a shifted Lévy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal.

The effect that weighted summands have on each other in approximations of $S={w_{1}}{S_{1}}+{w_{2}}{S_{2}}+\cdots +{w_{N}}{S_{N}}$ is investigated. Here, ${S_{i}}$’s are sums of integer-valued random variables, and ${w_{i}}$ denote weights, $i=1,\dots ,N$. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the ${S_{i}}$ has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.