Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T14:50:03.798Z Has data issue: false hasContentIssue false
  • English
  • Français

The correlated random walk with boundaries: A combinatorial solution

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

W. Böhm
W. Böhm*
Affiliation:
University of Economics, Vienna
*
∗Postal address: Institut fur Statistik, Abteilung fur Mathematische Methoden der Statistik, Augasse 2–6, A 1090 Wien, Austria. Email adress:boehm@isis.wu-wien.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.

Keywords

Correlated random walkcombinatorial solutionabsorbing boundaries

MSC classification

Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 470 - 479
Copyright
Copyright © by the Applied Probability Trust 2000 

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

Böhm, W. (1998). Multivariate Lagrange inversion and the maximum of a persistent random walk. J. Stat. Plann. Inference, to appear.Google Scholar
Brag, L. R. (1999). Trigonometric integrals and Hadamard products. Amer. Math. Monthly 106, 36–42.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley, New York.Google Scholar
Gessel, I. M., and Zeilberger, D. (1992). Random walk in Weyl chamber. Proc. American Math. Soc. 115, 27–31.CrossRefGoogle Scholar
Gillis, J. (1955). Correlated random walk. Proc. Camb. Phil. Soc. 51, 639–651.CrossRefGoogle Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. 4, 129–156.CrossRefGoogle Scholar
Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table of Integrals, Series and Products. Academic Press, Orlando, FL.Google Scholar
Henrici, P. (1977). Applied and Computational Complex Analysis. Vol. 2. John Wiley, New York.Google Scholar
Krattenthaler, C. (1997). The enumeration of lattice paths with respect to their number of turns. In Advances in Combinatorial Methods and Applications to Probability and Statistics, ed. Balakrishnan, N. Birkhäuser, Boston, MA.Google Scholar
Lal, R., and Bhat, U. N. (1989). Some explicit results for correlated random walks. J. Appl. Prob. 27, 757–766.Google Scholar
Mohan, C. (1955). The gambler's ruin problem with correlation. Biometrika 42, 486–493.Google Scholar
Mohanty, S. G. (1966). On a generalized two-coin tossing problem. Biometrische Z. 8, 266–272.Google Scholar
Mohanty, S. G. (1979). Lattice Path Counting and Applications. Academic Press, New York.Google Scholar
Renshaw, E., and Henderson, R. (1981). The correlated random walk. J. Appl. Prob. 18, 403–414.CrossRefGoogle Scholar
Weiss, G. H. (1994). Aspects and Applications of the Random Walk. North-Holland, Amsterdam.Google Scholar
Zhang, Y. L. (1992). Some problems on a one-dimensional correlated random walk with various types of barrier. J. Appl. Prob. 29, 196–201.Google Scholar