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

Affiliation(s)

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

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

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*.

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.

M. Vamvakari, "On Continuous Limiting Behaviour for the

References

[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. http:www.math.ufl.edu/ frank/qmaple/qmaple.html

[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. http:www.math.ufl.edu/ frank/qmaple/qmaple.html