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A monotone Sinai theorem. (English) Zbl 1359.37014

Let \(p=(p_i)_{i\in [N]}\) be any probability vector on the state set \([N]:= \{0,1,\dots,N-1\}\). For each \(n\geq 0\), define \( \mu_p(_m[i_0,\dots,i_n])=p_{i_0}p_{i_1}\dots p_{i_n}\), where \(i_0,i_1,\dots,i_n \in [N]\). Such a measure \(\mu_p\) is called a Bernoulli measure. Then \(B(p):= ([N]^{\mathbb Z},\mu_p,S)\) is called a Bernoulli shift on \(N\) symbols, where \(S\) is the left-shift given by \(S(y)_i=y_{i+1}\). The entropy of the Bernoulli shift \(B(p)\) is given by the positive number \(H(p):=-\sum^{n-1}_{i=0}p_i\log p_i\). Let \((X,\mu)\) be a probability space. If \(T:X\rightarrow X\) is a map such that \(\mu\circ T^{-1}=\mu\), then \((X,\mu,T)\) is a measure-preserving system. Ya. G. Sinai [Sov. Math., Dokl. 3, 1725–1729 (1963; Zbl 0205.13501); translation from Dokl. Akad. Nauk SSSR 147, 797–800 (1962)] has proved that if \((X,\mu,T)\) is a nonatomic invertible ergodic measure-preserving system of entropy \(h>0\), then it has any Bernoulli shift of any entropy \(h' \leq h\) as a factor. If \(p\) and \(q\) are two probability measures on \([N]:= \{0, 1,\dots,N - 1\}\) such that \(p\) stochastically dominates \(q\) (\(\sum^k_{i=0} p_i\leq \sum^k_{i=0}q_i\) for all \(0\leq k < N\)) and \(H(p) > H(q)\), does there exist a monotone factor map from \(B(p)\) to \(B(q)\)? In the present paper, the authors answer above question affirmatively. One of the main results in the paper is following:
Theorem 1. Let \(B(p)\) and \(B(q)\) be Bernoulli shifts with symbols in \([N]\) (where one allows the possibility that \(p\) and \(q\) give zero mass to some symbols). If the entropy of \(B(p)\) is strictly greater than that of \(B(q)\) and the measure \(p\) stochastically dominates \(q\), then \(B(q)\) is a monotone factor of \(B(p)\).
Secondly, the authors prove the following related result in the more restricted context of Bernoulli factors.
Theorem 2. Let \(B(p)\) and \(B(q)\) be Bernoulli shifts with symbols in \([N]\) (where one allows the possibility that \(p\) and \(q\) give zero mass to some symbols). Let \(R\) be any relation on \([N]\). If the entropy of \(B(p)\) is strictly greater than that of \(B(q)\), and the measure \(p\) \(R\)-dominates the measure \(q\), then there exists a factor \(\phi\) from \(B(p)\) to \(B(q)\) such that \((x_0,\phi(x_0))\in R\) for all \(x\in [N]^{\mathbb{Z}}\).
The paper is well written, the results are interesting in the sense that they suggest the existence of factor \(\phi\) from \(B(p)\) to \(B(q)\) satisfying the conditions of the theorems.

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
60G10 Stationary stochastic processes
60E15 Inequalities; stochastic orderings

Citations:

Zbl 0205.13501
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References:

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