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Axioms 2014, 3(2), 140-152; doi:10.3390/axioms3020140

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

Department of Informatics and Telematics, Harokopio University, 70 El. Venizelou str., Athens 17671, Greece
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Received: 11 November 2013 / Revised: 17 March 2014 / Accepted: 4 April 2014 / Published: 10 April 2014
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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. View Full-Text
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 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
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Kyriakoussis, A.; Vamvakari, M. Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions. Axioms 2014, 3, 140-152.

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