On Continuous Limiting Behaviour for the q(n)-Binomial Distribution with q(n)→1 as n→∞

Malvina Vamvakari

Department of Informatics and Telematics, Harokopio University of Athens, Athens, Greece.

DOI: 10.4236/am.2012.312A290                

Abstract

Recently, Kyriakoussis and Vamvakari [1] have established a q-analogue of the Stirling type for q-constant which have lead them to the proof of the pointwise convergence of the q-binomial distribution to a Stieltjes-Wigert continuous distribution. In the present article, assuming a sequence q(n) of n with q(n)→1 as n→∞, the study of the affect of this assumption to the q(n)-analogue of the Stirling type and to the asymptotic behaviour of the q(n)-Binomial distribution is presented. Specifically, a q(n) analogue of the Stirling type is provided which leads to the proof of deformed Gaussian limiting behaviour for the q(n)-Binomial distribution. Further, figures using the program MAPLE are presented, indicating the accuracy of the established distribution convergence even for moderate values of n.

Keywords

Stirling Formula; q(n) -Factorial Number of Order n; Saddle Point Method; q(n)-Binomial Distribution; Pointwise Convergence; Gauss Distribution

Conflicts of Interest

The authors declare no conflicts of interest.

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