Institute of Mathematical Statistics Collections

Absorption Time Distribution for an Asymmetric Random Walk

S. N. Ethier

Full-text: Open access

Abstract

Consider the random walk on the set of nonnegative integers that takes two steps to the left (just one step from state 1) with probability p[1/3,1) and one step to the right with probability 1p. State 0 is absorbing and the initial state is a fixed positive integer j0. Here we find the distribution of the absorption time. The absorption time is the duration of (or the number of coups in) the well-known Labouchere betting system. As a consequence of this, we obtain in the fair case (p=1/2) the asymptotic behavior of the Labouchere bettor’s conditional expected deficit after n coups, given that the system has not yet been completed.

Chapter information

Source
Stewart N. Ethier, Jin Feng and Richard H. Stockbridge, eds., Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 31-40

Dates
First available in Project Euclid: 28 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1233152933

Digital Object Identifier
doi:10.1214/074921708000000282

Zentralblatt MATH identifier
1170.60317

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Ethier, S. N. Absorption Time Distribution for an Asymmetric Random Walk. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 31--40, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000282. https://projecteuclid.org/euclid.imsc/1233152933


Export citation

References

  • [1] Aeppli, A. (1924). Zur Theorie verketteter Wahrscheinlichkeiten. Ph.D. thesis, University of Zurich.
  • [2] Barbier, É. (1887). Généralisation du problème résolu par M. J. Bertrand. Comptes Rendus des Séances de l’Académie des Sciences, Paris 105 407.
  • [3] Bertrand, J. (1887). Solution d’un problème. Comptes Rendus des Séances de l’Académie des Sciences, Paris 105 369.
  • [4] Downton, F. (1980). A note on Labouchere sequences. Journal of the Royal Statistical Society, Series A 143 363–366.
  • [5] Grimmett, G. R. and Stirzaker, D. R. (2001). One Thousand Exercises in Probability. Oxford University Press, Oxford.
  • [6] Koroljuk, V. S. (1955). On the discrepancy of empiric distributions for the case of two independent samples. Izvestiya Acad. Nauk SSSR. Ser. Mat. 19 81–96. Translated in IMS & AMS 4 105–122, 1963.
  • [7] Niederhausen, H. (2002). Catalan traffic at the beach. The Electronic Journal of Combinatorics 9 #R33.
  • [8] Takács, L. (1962). On the ballot theorems. In Advances in Combinatorial Methods and Applications to Probability and Statistics (N. Balakrishnan, ed.) 97–114. Birkhäuser, Boston.
  • [9] Thorold, A. L. (1913). The Life of Henry Labouchere. G. P. Putnam’s Sons, New York.