AM  Vol.3 No.12 A , December 2012
On Continuous Limiting Behaviour for the q(n)-Binomial Distribution with q(n)→1 as n→∞
Author(s) Malvina Vamvakari*

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.

Cite this paper
M. Vamvakari, "On Continuous Limiting Behaviour for the q(n)-Binomial Distribution with q(n)→1 as n→∞," Applied Mathematics, Vol. 3 No. 12, 2012, pp. 2101-2108. doi: 10.4236/am.2012.312A290.

[1]   A. Kyriakoussis and M. G. Vamvakari, “On a q-Analogue of the Stirling Formula and a Continuous Limiting Behaviour of the q-Binomial Distribution-Numerical Calculations,” Methodology and Computing in Applied Probability, 2011, pp. 1-27.

[2]   Ch. A. Charalambides, “Discrete q-Distributions on Bernoulli Trials with a Geometrically Varying Success Probability,” The Journal of Statistical Planning and Inference, Vol. 140, No. 9, 2010, pp. 2355-2383.

[3]   A. W. Kemp and C. D. Kemp, “Welson’s Dice Data Revisited,” The American Statistician, Vol. 45, No. 3, 1991, pp. 216-222.

[4]   A. W. Kemp, “Steady-State Markov Chain Models for Certain q-Confluent Hypergeometric Distributions,” Journal of Statistical Planning and Inference, Vol. 135, No. 1, 2005, pp. 107-120. doi:10.1016/j.jspi.2005.02.009

[5]   E. A. Bender, “Central and Local Limit Theorem Applied to Asymptotic Enumeration,” Journal of Combinatorial Theory Series A, Vol. 15, No. 1, 1973, pp. 91-111. doi:10.1016/0097-3165(73)90038-1

[6]   E. R. Canfield, “Central and Local Limit Theorems for the Coefficients of Polynomials of Binomial Type,” Journal of Combinatorial Theory, Series A, Vol. 23, No. 3, 1977, pp. 275-290. doi:10.1016/0097-3165(77)90019-X

[7]   P. Flajolet and M. Soria, “Gaussian Limiting Distributions for the Number of Components in Combinatorial Structures,” Journal of Combinatorial Theory Series A, Vol. 53, No. 2, 1990, pp. 165-182. doi:10.1016/0097-3165(90)90056-3

[8]   A. M. Odlyzko, “Handbook of Combinatorics,” In: R. L. Graham, M. Grotschel and L. Lovász, Eds., Asymptotic Enumeration Methods, Elsevier Science Publishers, Amsterdam, 1995, pp. 1063-1229.?

[9]   L. M. Kirousis, Y. C. Stamatiou and M. Vamvakari, “Upper Bounds and Asymptotics for the q-Binomial Coefficients,” Studies in Applied Mathematics, Vol. 107, No. 1, 2001, pp. 43-62. doi:10.1111/1467-9590.1071177

[10]   A. Kyriakoussis and M. Vamvakari, “On Asymptotics for the Signless Noncentral q-Stirling Numbers of the First Kind,” Studies in Applied Mathematics, Vol. 117, No. 3, 2006, pp. 191-213. doi:10.1111/j.1467-9590.2006.00352.x

[11]   Ch. A. Charalambides and A. Kyriakoussis, “An Asymptotic Formula for the Exponential Polynomials and a Central Limit Theorem for Their Coefficients,” Discrete Mathematics, Vol. 54, No. 3, 1985, pp. 259-270. doi:10.1016/0012-365X(85)90110-4

[12]   P. Flajolet and R. Sedgewick, “Analytic Combinatorics, Chapter V: Application of Rational and Meromorphic Asymptotics, Chapter VIII: Saddle Point Asymptotics,” Cambridge University Press, Cambridge, 2009.

[13]   F. Garvan, “The Maple QSERIES Package,” 2010. frank/qmaple/qmaple.html