New integrated view at partial-sums distributions

Abstract

ABSTRACT. Partial-sums discrete probability distributions occurred in description of many stochastic models. They were used also as a tool for creating new distributions, or as a link between known distributions. It is shown in this paper that every discrete distribution with only non-zero probabilities is a partial-sums distribution, and, moreover, that it has infinitely many parent distributions. The paper generalizes and unifies the concept of partial-sums distribution. Besides, it generalizes some risk models in insurance and revises some approaches to mathematical modelling in quantitative linguistics.

On a Parametrization of Partial-Sums Discrete Probability Distributions

“…the necessary condition (19) is not satisfied, which means that the logarithmic distribution belongs to the sensitive family.…”

“…Again, it is easy to see that the necessary condition (19) is satisfied by the geometric distribution (Example 2), whereas it is not true for the Poisson distribution (Example 1).…”

“…Depending on the choice of function g(j) from ( 4), some constraints on the parent distribution may be necessary, e.g., the mean of the parent distribution must be finite under (1), see [18]. In [19], it was shown that all pairs of discrete distributions are connected by a summation (4), i.e., for any pair of discrete probability distributions {P * j } ∞ j=0 and {P j } ∞ j=0 there is a partial summation (4) such that {P * j } ∞ j=0 is the parent of {P j } ∞ j=0 . Limit properties of partial-sums distributions were derived in [20][21][22] .…”

On the limit behaviour of finite-support bivariate discrete probability distributions under iterated partial summations

“…Research on partial-sums distributions is almost exclusively dedicated to univariate distributions (see Wimmer and Mačutek (2012), and references therein). The only note on the bivariate (and r-variate) partial-sums distributions can be found in Kotz and Johnson (1991), who more or less restrict themselves to a suggestion to study multivariate cases.…”