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By Manfred Denker

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7) Theorem of Perron-Frobenius; The positive matrix R = (rij) has a positive eigenvalue r such that no eigenvalue of R has absolute value > r. One has min i ~ j < r < max rij --- i ~ j rij . 43 To this dominant eigenvalue r there correspond positive left (row) and right (column) eigenvectors. If R is irreducible, the eigenvalue r is simple, and the corresponding eigenvectors are strictly positive. For a proof of this theorem, we refer to ~97 , chapter 13~. 8) Definition: An sxs-matrix P = (pij) is said to be a stochastic matrix if p..

A N) in S X let m[al .... ,aN] block in x at the place m. ,aN) tered cylinder the set of all x E S 2 os length N based on the at the place m. ,a2M+l]. In the literature a block is sometimes called a w o r d and a cylinder is called a thin cylinder. Remark that if x is an element of some centered of length 2M + I, then the i-th coordinate cylinder of x is specified for all Ill ~ M. If x (n) E S, x E S z and x (n) ~x, then for each i E Z there is an N(i) such that (x(n)~-- = x i for all n > N(i).

1 0 0 . 9 of period s is given by O" 0 . 9 9 . 15) says that this is (possibly after relabelling the states in S) the form of a general periodic matrix, with the 1's replaced by stochastic sub-matrices, and the O's by submatrices consisting of O's. SThen lim p ~ J exists and is equal to pj > 0 for all i,j E S. For a proof, we refer to the standard texts of probability theory. 17) Proposition: P is aperiodic iff there exists an N such that p ~10 > 0 for all n > N and all i,j E X. 18) Proposition: The following conditions (a) U~p is weakly mixing; (b) Uw P is strongly mixing; (c) P is aperiodic.

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