Modern Stochastics: Theory and Applications

The purpose of this paper is to explore two probability distributions originating from the Kies distribution defined on an arbitrary domain. The first one describes the minimum of several Kies random variables whereas the second one is for their maximum – they are named min- and max-Kies, respectively. The properties of the min-Kies distribution are studied in details, and later some duality arguments are used to examine the max variant. Also the saturations in the Hausdorff sense are investigated. Some numerical experiments are provided.

The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.