Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions

Abstract

From Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of this family and we establish a continuous limiting behavior for the members I and II, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we initially give the q-factorial moments and the usual moments for the family of confluent q-Chu- Vandermonde distributions and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over/underdispersed (O/U) distribution. Note that in this article we prove the convergence of a family of discrete distributions to a continuous distribution which is not of a Gaussian type.

Keywords

• stirling asymptotic formula
• q-factorial number of order n
• confluent q-Chu-Vandermonde distributions
• q-factorial moments
• modified q-Bessel distribution
• generalized q-negative Binomial distribution
• Over/Underdispersed (O/U) distribution
• pointwise convergence
• continuous Stieltjes-Wigert distribution