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3 results

  • Decidability of α (pk) > 0 for some k

  • Richard W. Madsen
  • Journal:
    Journal of Applied Probability / Volume 12 / Issue 2 / June 1975
    Published online by Cambridge University Press:
    14 July 2016, pp. 333-340
    Print publication:
    June 1975

  • Paz (1963), (1971) has shown that for an n × n stochastic matrix P the property that some power of P has a positive ergodic coefficient is decidable. In particular he shows that α (Pk) > 0 for some k, only if it is positive for k = ½n(n – 1). However he states that it is not known whether this bound is sharp. In this paper a sharp bound is given, namely k =½(n – 1)2 + ½ or ½(n − 1)2 + 1 depending on whether n is even or odd. The proof of this is based on some results from number theory.

  • Positive columns for stochastic matrices

  • Dean Isaacson, Richard Madsen
  • Journal:
    Journal of Applied Probability / Volume 11 / Issue 4 / December 1974
    Published online by Cambridge University Press:
    14 July 2016, pp. 829-835
    Print publication:
    December 1974

  • If an n × n stochastic matrix has a column with no zeros, one can immediately conclude that the chain is ergodic and the state corresponding to that column is persistent and aperiodic. In this paper it is shown that it is decidable whether or not some power of a finite stochastic matrix has a positive column. Some problems regarding positive columns in infinite stochastic matrices are also considered.