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2 results

  • Markov Chains in Many Dimensions

  • Part of
  • Dimitris N. Politis
  • Journal:
    Advances in Applied Probability / Volume 26 / Issue 3 / September 1994
    Published online by Cambridge University Press:
    01 July 2016, pp. 756-774
    Print publication:
    September 1994

  • A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

  • Longest common subsequences of two random sequences

  • Vacláv Chvátal, David Sankoff
  • Journal:
    Journal of Applied Probability / Volume 12 / Issue 2 / June 1975
    Published online by Cambridge University Press:
    14 July 2016, pp. 306-315
    Print publication:
    June 1975

  • Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n–1f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).