A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. Using the introduced transform and Stein’s method, we investigate the rate of convergence in the Rényi theorem for the distributions of geometric sums of independent random variables with identical nonzero means and finite second moments without any constraints on their supports. We derive an upper bound for the Kantorovich distance between the normalized geometric random sum and the exponential distribution which has exact order of smallness as the expectation of the geometric number of summands tends to infinity. Moreover, we introduce the so-called asymptotically best constant and present its lower bound yielding the one for the Kantorovich distance under consideration. As a concluding remark, we provide an extension of the obtained estimates of the accuracy of the exponential approximation to non-geometric random sums of independent random variables with non-identical nonzero means.

On chi-square type distributions with geometric degrees of freedom in relation to geometric sums

The chi-square distribution with degrees of freedom has an important role in probability, statistics and various applied fields as a special probability distribution. This paper concerns the relations between geometric random sums and chi-square type distributions whose degrees of freedom are geometric random variables. Some characterizations of chi-square type random variables with geometric degrees of freedom are calculated. Moreover, several weak limit theorems for the sequences of chi-square type random variables with geometric random degrees of freedom are established via asymptotic behaviors of normalized geometric random sums.

The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution

The geometric sums have been arisen from the necessity to resolve practical problems in ruin prob- ability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of characteristic functions, the necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution are shown. In special case, obtained result is a weak limit theorem for geometric sums of independent and identically distributed random variables which has been well-known as the second central limit theorem. Furthermore, based on the obtained results of this paper, the analogous results for the domains of geometric attraction of exponential distribution and Linnik distribution can be established. More generally, we may extend results to the domain of geometric attraction of geometrically strictly stable distributions.

Exploring the Mathematics of Bouncing Balls

Simple equipment—stopwatches, tape measures, and golf balls—allows students to explore geometric sums and quadratic equations and analyze experimental data.