Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T08:47:37.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluctuation Theory for the Ehrenfest urn

Published online by Cambridge University Press:  01 July 2016

N. H. Bingham
N. H. Bingham*
Affiliation:
Royal Holloway and Bedford New College, London
*
∗Postal address: Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey TW20 0EX, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The Ehrenfest urn model with d balls, or alternatively random walk on the unit cube in d dimensions, is considered in discrete and continuous time, together with related models. Attention is focused on the fluctuation theory of the model—behaviour on unusual states—and in particular on first passage to the opposite vertex. Applications to statistical mechanics, reliability theory and genetics are surveyed, and some new results are obtained.

Keywords

RANDOM WALKUNIT CUBE IN D DIMENSIONSGROUP REPRESENTATIONSGELFAND PAIRSFIRST-PASSAGE TIMEGENETICSRELIABILITY THEORYSTATISTICAL MECHANICS
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 598 - 611
Copyright
Copyright © Applied Probability Trust 1991 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. (1982a) Some inequalities for reversible Markov chains. J. London Math. Soc. 25, 564–576.Google Scholar
Aldous, D. (1982b) Markov chains with almost exponential hitting times. Stoch. Proc. Appl. 13, 305–310.Google Scholar
Aldous, D. (1983a) Random walks on finite groups and rapidly mixing Markov chains. Lecture Notes in Mathematics 986, 243–297, Springer-Verlag, Berlin.Google Scholar
Aldous, D. (1983b) Minimization algorithms and random walks on the d-cube. Ann. Prob. 11, 403–413.Google Scholar
Aldous, D. and Diaconis, P. (1986) Shuffling cards and stopping times. Amer. Math. Monthly 93, 333–348.Google Scholar
Barlow, M. T. and Perkins, E. A. (1988) Brownian motion on the Sierpinski gasket. Prob. Theory Rel. Fields 79, 543–623.Google Scholar
Bellman, R. and Harris, T. E. (1951) Recurrence times for the Ehrenfest model. Pacific J. Math. 1, 179–193.Google Scholar
Callaert, H. and Keilson, J. (1973) On exponential ergodicity and spectral structure for birth-death processes, I, II. Stoch. Proc. Appl. 1, 187–216, 217–235.Google Scholar
Coxeter, H. S. A. (1973) Regular Polytopes , 3rd edn, Dover, New York.Google Scholar
Diaconis, P. (1988) Group Representations in Probability and Statistics. Lecture Notes—Monographs 11, Inst. Math. Statistics.Google Scholar
Diaconis, P. and Smith, L. (1988) Fluctuation theory for Gelfand pairs. Unpublished manuscript.Google Scholar
Donnelly, K. P. (1983) The probability that related individuals share some section of genome identical by descent. Theoret. Popn Biol. 23, 34–63.Google Scholar
Doob, J. L. (1942) The Brownian motion and stochastic equations. Ann. Math. 43, 351–369.Google Scholar
Ehrenfest, P. and Ehrenfest, T. (1959) The Conceptual Foundations of the Statistical Approach in Mechanics (transl. Moravcsik, M. J.: German original, 1912). Cornell University Press, Ithaca, NY.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Volume I, 2nd edn. Wiley, New York.Google Scholar
Giorno, V., Lánsky, P. and Nobile, A. G. (1988) Diffusion approximations and first-passage-time problems for a model neuron. III: A birth-and-death process approach. Biol. Cybernet. 58, 387–404.Google Scholar
Johnson, N. L. and Kotz, S. (1977) Urn Models and their Applications: an Approach to Modern Discrete Probability. Wiley, New York.Google Scholar
Kac, M. (1947) Random walk and the theory of Brownian motion. Amer. Math. Monthly 54, 369–391.Google Scholar
Kakosyan, A. V., Klebanov, L. B. and Melamed, J. A. (1984) Characterisation of Distributions by the Method of Intensively Monotone Operators. Lecture Notes in Mathematics 1088, Springer-Verlag, Berlin.Google Scholar
Karlin, S. and Mcgregor, J. (1957) The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
Karlin, S. and Mcgregor, J. (1961) The Hahn polynomials, formulas and an application. Scripta Math. 23, 33–46.Google Scholar
Karlin, S. and Mcgregor, J. (1964) On some stochastic models in genetics. Conference on Stochastic Models in Medicine and Biology , pp. 245–279, University of Wisconsin Press, Madison, WI.Google Scholar
Karlin, S. and Mcgregor, J. (1965) Ehrenfest urn models. J. Appl. Prob. 2, 352–376.Google Scholar
Keilson, J. (1979) Markov Chain Models—Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kemperman, J. H. B. (1961) The Passage Problem for a Stationary Markov Chain. University of Chicago Press, Chicago, IL.Google Scholar
Kent, J. T. (1978) Time-reversible diffusions. Adv. Appl. Prob. 10, 819–835 (Acknowledgement of priority, ibid. 11 (1979), 888).Google Scholar
Kent, J. T. (1980) Eigenvalue expansions for diffusion hitting times. Z. Wahrscheinlichkeitsth 52, 309–319.Google Scholar
Kent, J. T. (1982) The spectral decomposition of a diffusion hitting time. Ann. Prob. 11, 207–219.Google Scholar
Kent, J. T. and Longford, N. T. (1983) An eigenvalue decomposition for first hitting times in random walks. Z. Wahrscheinlichkeitsth. 63, 71–84.Google Scholar
Letac, G. (1981) Problèmes classiques de probabilité sur un couple de Gelfand. In Analytical Methods in Probability Theory , Lecture Notes in Mathematics 861, 93–120, Springer-Verlag, Berlin.Google Scholar
Letac, G. (1982) Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov. Adv. Appl. Prob. 14, 272–294.Google Scholar
Letac, G. and Takács, L. (1979) Random walk on the m-dimensional cube. J. reine angew. Math. 310, 187–195.Google Scholar
Letac, G. and Takács, L. (1980a) Random walks on a 600-cell. SIAM J. Algeb. Discrete Methods 1, 114–120.Google Scholar
Letac, G. and Takács, L. (1980b) Random walk on a dodecahedron. J. Appl. Prob. 17, 373–384.Google Scholar
Moran, P. A. P. (1968) An Introduction to Probability Theory. Oxford University Press (2nd edn 1984).Google Scholar
Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985a) Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.Google Scholar
Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985b) Exponential trends of first-passage-time densities for a class of diffusions with steady-state distributions. J. Appl. Prob. 22, 611–618.Google Scholar
Pitman, J. W. and Rogers, L. C. G. (1981) Markov functions. Ann. Prob. 9, 573–582.Google Scholar
Renyi, A. (1956) A characterisation of Poisson processes (Hungarian; Russian and English summaries). Magyar Tud. AMKIK 1, 519–527. (English translation in Selected Papers of Alfred Rényi, Volume 1, 622–627, Akadémiai Kiadó, Budapest, 1976).Google Scholar
Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics 14. Springer-Verlag, Berlin.Google Scholar
Ricciardi, L. M. (1982) Diffusion approximations and computational problems for single neuron's activity. Lecture Notes in Biomathematics 45, pp. 142–154, Springer-Verlag, Berlin.Google Scholar
Ricciardi, L. M. (1985) Diffusion approximations and first-passage-time problems in population biology and neurobiology. Lecture Notes in Biomathematics 57, 455–468, Springer Verlag, Berlin.Google Scholar
Severo, N. C. (1970) A note on the Ehrenfest multi-urn model. J. Appl. Prob. 7, 444–445.Google Scholar
Siegert, A. J. F. (1949) On the approach to statistical equilibrium. Phys. Rev. 71, 1708–1714.Google Scholar
Siegert, A. J. F. (1951) On the first-passage-time probability problem. Phys. Rev. 81, 617–623.CrossRefGoogle Scholar
Soto-Andrade, J. (1988) Métodos geométricos en teoría de representaciones de grupos finitos . Santiago, Chile.Google Scholar
Stanton, D. (1984) Orthogonal polynomials and Chevalley groups. In Special Functions: Group-Theoretical Aspects and Applications , ed. Askey, R. et al., pp. 87–92, Reidel, Dordrecht.Google Scholar
Takács, L. (1979) On an urn problem of Paul and Tatiana Ehrenfest. Math. Proc. Camb. Phil. Soc. 86, 127–130.Google Scholar
Takács, L. (1981) Random flights on regular polytopes. SIAM J. Algeb. Discrete Methods 2, 153–171.Google Scholar
Takács, L. (1982) Random walks on groups. Linear Algeb. Appl. 43, 49–67.Google Scholar
Takács, L. (1983) Random walk on a finite group. Acta Sci. Math. (Szeged) 45, 395–408.Google Scholar
Takács, L. (1984) Random flights on regular graphs. Adv. Appl. Prob. 16, 618–637.Google Scholar
Takács, L. (1986) Harmonic analysis on Schur algebras and its applications to the theory of probability. In Probability Theory and Harmonic Analysis , ed. Chao, J. A. and Woyczynski, W. A., pp. 227–283, Marcel Dekker, New York.Google Scholar
Vincze, I. (1964) Über das Ehrenfestsche Modell der Wärmeübertragung. Arch. Math. 15, 394–400.Google Scholar
Walsh, J. B. (1981) A stochastic model of neural response. Adv. Appl. Prob. 13, 231–281.Google Scholar