Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T11:29:22.344Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit property under a modified Ehrenfest urn design

Part of: Markov processes Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Yung-Pin Chen
Yung-Pin Chen*
Affiliation:
Lewis & Clark College
*
∗Postal address: Department of Mathematical Sciences, Lewis & Clark College, Portland, OR 97219, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a stochastic process in a modified Ehrenfest urn model. The modification prescribes there to be a minimum number of balls in each urn, and the process records the differences between treatment assignments under a sampling scheme implemented with this modified Ehrenfest urn model. In contrast to the result that the difference process forms a Markov chain and converges to a stationary distribution under the Ehrenfest urn model, the corresponding process under this modified Ehrenfest urn design satisfies the central limit property. We prove this asymptotic normality property using a central limit theorem for dependent random variables, renewal theory, and two Kolmogorov-type maximal inequalities.

Keywords

Ehrenfest urn modelMarkov chainrenewal theorycentral limit propertymaximal inequality

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 409 - 420
Copyright
© Applied Probability Trust 2006 

References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bingham, N. H. (1991). Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.Google Scholar
Blackwell, D. and Hodges, J. L. Jr. (1957). Design for the control of selection bias. Ann. Math. Statist. 28, 449–460.Google Scholar
Chen, Y.-P. (2000). Which design is better? Ehrenfest urn versus biased coin. Adv. App. Prob. 32, 738–749.Google Scholar
Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics, Hayward, CA.Google Scholar
Ehrenfest, P. and Ehrenfest, T. (1907). Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem. Phys. Z. 8, 311–314.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Hoel, P. G., Port, S. C. and Stone, C. J. (1972). Introduction to Stochastic Processes. Houghton Mifflin, Boston, MA.Google Scholar
Ibragimov, I. (1962). Some limit theorems for stationary processes. Theor. Prob. Appl. 7, 349–382.CrossRefGoogle Scholar
Ibragimov, I. (1963). A central limit theorem for a class of dependent random variables. Theor. Prob. Appl. 8, 83–89.Google Scholar
Karlin, S. and McGregor, J. (1959). Random walks. Illinois J. Math. 3, 66–81.Google Scholar
Karlin, S. and Taylor, M. H. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Palacios, J. L. (1993). Fluctuation theory for the Ehrenfest urn via electric networks. Adv. Appl. Prob. 25, 472–476.Google Scholar
Palacios, J. L. (1994). Another look at the Ehrenfest urn via electric networks. Adv. Appl. Prob. 26, 820–824.Google Scholar
Schach, S. (1971). Weak convergence results for a class of multivariate Markov processes. Ann. Math. Statist. 42, 451–465.Google Scholar
Serfling, R. J. (1968). Contributions to central limit theory for dependent variables. Ann. Math. Statist. 39, 1158–1175.Google Scholar
Van Beek, K. W. H. and Stam, A. J. (1987). A variant of the Ehrenfest model. Adv. App. Prob. 19, 995–996.CrossRefGoogle Scholar