ROTATIONAL REPRESENTATIONS OF TRANSITION MATRIX FUNCTIONS

• 成果类型: Article
• 署名作者: KALPAZIDOU, S
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988726
• 发表日期: 1994
• 页码: 703-712
• 关键词: stochastic matrices; markov-chains

摘要

Let P(h) = (p(ij)(h),i,j= 1,2, ..., n),h greater-than-or-equal-to 0,n greater-than-or-equal-to 1, be a transition matrix function defining an irreducible recurrent continuous parameter Markov process. Let (S(i), i = 1, 2, ..., n) be a partition of the circle into sets S(i) each consisting of a finite union of arcs A(kl). Let f(t) be a rotation of length t of the circle, and denote Lebesgue measure by lambda. We generalize and prove for the transition matrix function P(h) a theorem of Cohen (n = 2) and Alpern (n > 2) asserting that every recurrent stochastic n x n matrix P is given by (*) p(ij) = (lambda(S(i) and f(t)-1(S(j)))/lambda(S(i)), for some choice of rotation ft and partition {S(i)}. We prove the existence of a continuous map PHI from the space of n x n irreducible stochastic matrices into n-partitions of [0, 1), such that every domain matrix P is represented by (*) with {S(i)} = PHI(P) and t = 1/n!. Furthermore, the representing process (f(t), {S(i)}) has not only the same transition probabilities but also the same probabilistic cycle distribution as the Markov process based on P.