ON THE ROTATIONAL DIMENSION OF STOCHASTIC MATRICES

• 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/1176988298
• 发表日期: 1995
• 页码: 966-975
• 关键词: markov-chains; REPRESENTATION

摘要

Let (S-i, i = 1, 2,..., n), n > 1, be a partition of the circle into sets S-i each consisting of union of delta(i) < infinity, arcs A(kl). Let f(t) be a rotation of length t of the circle and denote Lebesgue measure by lambda. Then every recurrent stochastic matrix P on S = {1,..., n} is given according to a theorem of Cohen (n = 2), Alpern and Kalpazidou (n greater than or equal to 2) by p(ij) = lambda S-i boolean AND f(t)(-1)(S-j))/lambda(Si) for some choice of rotation f(t) and partition L = {S-i}. The number delta(L) = max(i) delta(i) is called the length of description of the partition L. Then it turns out that the minimal value of delta(L), when L varies, characterizes the matrix P. We call this value the rotational dimension of P. We prove that for certain recurrent n X n stochastic matrices the rotational dimension is provided by the number of Betti circuits of the graph of P. One preliminary result shows that there are recurrent n x n stochastic matrices which admit minimal positive circuit decompositions in terms of the Betti circuits of their graph. Finally, a generalization of the rotational dimension for the transition matrix functions is also given.