Random Walk

  • Bert Fristedt
  • Lawrence Gray
Part of the Probability and its Applications book series (PA)


In this chapter, we will study certain sequences of random variables, known as ‘random walks’. These are defined in terms of sums of independent identically distributed random variables. Important in the study of random walks (and of more general random sequences) are ‘filtrations’ and ‘stopping times’. A filtration is a sequence of σ-fields representing the information available at various stages of an experiment. A stopping time is a ℤ̄+ -valued random variable whose value may be regarded as the time at which an experiment is to be terminated. In applications, such as gambling theory, important stopping times are the time at which a random walk reaches a certain goal and the time at which it returns to its original position. These will be treated in the latter part of the chapter for several special random walks.


Random Walk Random Sequence Return Time Prob Ability Simple Random Walk 
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  1. Athreya, K. B. and Ney, P. E., Branching Processes, Springer-Verlag, New York, 1972.CrossRefMATHGoogle Scholar
  2. Blumenthal, R. M. and Getoor, R. K., Markov Processes and Potential Theory, Academic Press, New York, 1968.MATHGoogle Scholar
  3. Chen, Mu Fa, From Markov Chains to Non-Equilibrium Particle Systems, World Scientific, Singapore, 1992.CrossRefMATHGoogle Scholar
  4. Chow, Y. S. and Robbins, Herbert and Siegmund, David, Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston, 1971.MATHGoogle Scholar
  5. Chung, Kai Lai, Markov Chains with Stationary Transition Probabilities, Second Edition, Springer-Verlag, New York, 1967.MATHGoogle Scholar
  6. Chung, Kai Lai, Lectures on Boundary Theory for Markov Chains, Princeton University Press, Princeton, New Jersey, 1970.MATHGoogle Scholar
  7. Dellacherie, Claude and Meyer, Paul-André, Probabilities and Potential C (translated from French), Elsevier Science Publishers B. V., Amsterdam, 1988.MATHGoogle Scholar
  8. Dellacherie, Claude and Meyer, Paul-André, Probabilités et Potential (Théorie du potentiel associée à une résolvante, Théorie des processus de Markov), Hermann, Paris, 1987.MATHGoogle Scholar
  9. Doob, J. L., Classical Potential Theory and its Probabilistic Counterpart, Springer-Verlag, New York, 1984.CrossRefMATHGoogle Scholar
  10. Doyle, Peter G. and Snell, J. Laurie, Random Walks and Electric Networks, Mathematical Association of America, 1984.MATHGoogle Scholar
  11. Dynkin, E. B., Markov Processes, Vol. I, Academic Press, New York, 1965.Google Scholar
  12. Dynkin, E. B., Markov Processes, Vol. II, Academic Press, New York, 1965.Google Scholar
  13. Dynkin, Evgenii B. and Yushkevich, Alexsandr A., Markov Processes: Theorems and Problems, Plenum Press, New York, 1969.CrossRefGoogle Scholar
  14. Edgar, G. A. and Sucheston, Louis, Stopping Times and Directed Processes (Encyclopedia of Mathematics and Its Applications, Vol. 47), Cambridge University Press, Cambridge, 1992.CrossRefGoogle Scholar
  15. Ethier, Stewart N. and Kurtz, Thomas G., Markov Processes: Characterization and Convergence, John Wiley & Sons, New York, 1986.CrossRefMATHGoogle Scholar
  16. Preedman, David, Approximating Countable Markov Chains, Holden-Day, San Francisco, 1971.Google Scholar
  17. Preedman, David, Markov Chains, Holden-Day, San Francisco, 1971.Google Scholar
  18. Harris, Theodore E., The Theory of Branching Processes, Dover, New York, 1989.Google Scholar
  19. Hughes, Barry D., Random Walks and Random Environments, Vol. 1: Random Walks, Clarendon Press, Oxford, 1995.MATHGoogle Scholar
  20. Hughes, Barry D., Random Walks and Random Environments, Vol. 2: Random Environments, Clarendon Press, Oxford, 1996.MATHGoogle Scholar
  21. Iosifescu, Marius, Finite Markov Processes and Their Applications, John Wiley & Sons, Chichester, 1980.MATHGoogle Scholar
  22. Kalashnikov, Vladimir V., Topics on Regenerative Processes, CRC Press, Boca Raton, Florida, 1994.MATHGoogle Scholar
  23. Kemeny, John G. and Snell, J. Laurie and Knapp, Anthony W., Denumerable Markov Chains, D. van Nostrand, Princeton, New Jersey, 1966.MATHGoogle Scholar
  24. Kingman, J. F. C., Regenerative Phenomena, John Wiley & Sons, London, 1972.MATHGoogle Scholar
  25. Lawler, Gregory F., Intersections of Random Walks, Birkhäuser, Boston, 1991.CrossRefMATHGoogle Scholar
  26. Maisonneuve, Bernard, Systèmes Régénératifs (Astérique, Vol. 15), Société Mathématique de France, Paris, 1974.Google Scholar
  27. Révész, Pál, Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990.CrossRefMATHGoogle Scholar
  28. Sharpe, Michael, General Theory of Markov Processes, Academic Press, Boston, 1988.MATHGoogle Scholar
  29. Spitzer, Frank, Principles of Random Walk, Second Edition, Springer-Verlag, New York, 1976.CrossRefMATHGoogle Scholar
  30. Tackás, Lajos, Combinatorial Methods in the Theory of Stochastic Processes, John Wiley & Sons, New York, 1967.Google Scholar
  31. Yang, Xiang-qun, The Construction Theory of Denumerable Markov Processes, John Wiley & Sons, Chichester, 1990.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Bert Fristedt
    • 1
  • Lawrence Gray
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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