Advertisement

Cycle Representations of Markov Processes: An Application to Rotational Partitions

  • S. Kalpazidou
Part of the Trends in Mathematics book series (TM)

Abstract

The cycle formula asserts that any finite-order recurrent stochastic matrix P is a linear combination of matrices J c = (J c(i, j)) associated with the cycles c of the graph of P, and defined as follows: J c (i, ii) = 1 or 0, according to whether i, j are consecutive points of c or not.

In the present paper we investigate, by using the cycle formula, the asymptotic behavior of the sequence (t, m S), t ⩾ 0, of rotational representations associated to the powers P m, m = 0, 1, 2,…, of an irreducible stochastic matrix on a finite set S = {1,…, n}, n ⩾ 2. In particular, we give a criterion on the rotational partitions m S, m = 0, 1,…, for the sequence P m m to be convergent. A pair (t, S) is a rotational representation of P = (p ij, i, j = 1,…, n) if S is a partition of [0, 1) into n sets S 1,…, S n, each of positive Lebesgue measure and consisting of a finite union of arcs, such that Pij = . Here f t is the A-preserving transformation of [0, 1) onto itself defined by f t (x) = (x + t)(mod 1), and A denotes Lebesgue measure

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Aldous, P. Diaconis, and M. J. Steele, Discrete Probability and Algorithms, Springer-Verlag, New York, 1995.zbMATHCrossRefGoogle Scholar
  2. [2]
    S. Alpern, Rotational representations of stochastic matrices, Ann. Probability 11 (3), (1983), 789–794.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    J. E. Cohen, A geometric representation of stochastic matrices; theorem and conjecture, Ann. Probability 9 (1981), 899–901.zbMATHCrossRefGoogle Scholar
  4. [4]
    Y. Derriennic, Ergodic problems on random walks in random environment, in Selected Talks Delivered at the Department of Mathematics of the Aristotle University, S. Kalpazidou ed., Aristotle University Press, Thessaloniki, 1993.Google Scholar
  5. [5]
    J. Haigh, Rotational representation of stochastic matrices, Ann. Probability 13 (1985), 1024–1027.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    S. Kalpazidou, Rotational representations of transition matrix functions, Ann. Probability 22 (2), (1994), 703–712.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    S. Kalpazidou, On the rotational dimension of stochastic matrices, Ann. Probability 23 (2), (1995), 966–975.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S. Kalpazidou, Cycle Representations of Markov Processes, Springer-Verlag, New York, 1995.zbMATHGoogle Scholar
  9. [9]
    S. Kalpazidou and J.E. Cohen, Orthogonal cycle transforms of stochastic matrices, Circ. Sys. Sig. Proc., 16 (2), (1997), 363–374.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    S. Kalpazidou and N. Kassimatis, Markov chains in Banach spaces on cycles, Circ. Sys. Sig. Proc., to appear.Google Scholar
  11. [11]
    S. Kalpazidou, From network problem to cycle processes, Proceedings of the 2nd World Congress of Nonlinear Analysts, Athens, 1996, Elsevier, part 4, 2041–2049.Google Scholar
  12. [12]
    Qian Minping and Qian Min, Circulation for recurrent Markov chain, Z. Wahrsch. Verw. Gebiete 59 (1982), 203–210.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    P. M. Soardi, Potential Theory on Infinite Networks, Lecture Notes in Mathematics, Springer-Verlag, New York, 1994.Google Scholar
  14. [14]
    P. Del Tio Rodriguez and M. C. Valsero Blanco, A characterization of reversible Markov chains by a rotational representation, Ann. Probability 19 (2), (1991), 605–608.zbMATHCrossRefGoogle Scholar
  15. [15]
    W. Woess, Random walks on infinite graphs and groups-A survey on selected topics, Bull London Math. Soc. 26 (1994), 1–60.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    A. H. Zemanian, Infinite Electrical Networks, Cambridge University Press, Cambridge, 1991.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • S. Kalpazidou
    • 1
  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations