The Forward and Backward Rotational Decompositions of Markov Chains

Kalpazidou, S.; Tzouvaras, L.

doi:10.1081/sap-100002018

Cited by 2 publications

(2 citation statements)

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“…Cohen [8] considers the following finite dimensional variant of Theorem 2.2, the so-called rotational representations of P: Given an n × n stochastic matrix (p(i, j) : 1 ≤ i, j ≤ n), find a circle rotation τ and a circle partition P consisting of intervals satisfying (2.1). See also the work of Alpern [2], Haigh [12], Kalpazidou [15], Kalpazidou-Tzouvaras [14] and Alpern-Prasad [6] for related developments on rotational representations of stochastic matrices and cycle decompositions of Markov chains.…”

Section: Theorem 22 (Cmc) Let τ Be An Aperiodic µ-Preserving Automomentioning

confidence: 99%

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

Alpern

Prasad

2008

Proc. Amer. Math. Soc.

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We dedicate this paper to the memory of Shizuo Kakutani. We miss his kind manner, gentle presence and keen insight.Abstract. We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation τ of a Lebesgue probability space (X, A, µ) are equivalent. One theorem asserts that the conjugates of τ are dense in the uniform topology on the space of automorphisms. The other two results assert the existence of a partition of the space X with special properties. One partition result shows that given a mixing Markov chain, there is a code (i.e., a partition of the space) so that the itinerary process given by τ and the partition has the distribution of the given Markov Chain. The other partition result is a generalization of the Rokhlin Lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance. Thus the first two results are equivalent to this strengthening of Rokhlin's Lemma.

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Section: Theorem 22 (Cmc) Let τ Be An Aperiodic µ-Preserving Automomentioning

confidence: 99%

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

Alpern

Prasad

2008

Proc. Amer. Math. Soc.

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“…He called a solution f; S of (1) of this type a rotational representation of P: Cohen showed that such representations always exist for 2 2 irreducible stochastic matrices and conjectured that this result could be extended to n n matrices. The subsequent results of Alpern [3], Haigh [9], Rodrigues del Tio and Valsero Blanco [20], and Kalpazidou [11][12] [15], established and extended Cohen's conjecture in various ways. The purpose of this paper is to show that the cycle decomposition techniques used in [3] to establish Cohen's conjecture, together with the multitower constructions of [1] [3], can be modi…ed to give short elementary proofs of results on representations by automorphisms of various types, including the well known Coding Theorem (Theorem 9).…”

Section: Introductionmentioning

confidence: 99%

Rotational (and Other) Representations of Stochastic Matrices

Alpern

Prasad

2007

Stochastic Analysis and Applications

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Joel E. Cohen (1981) conjectured that any stochastic matrix P = fp i;j g could be represented by some circle rotation f in the following sense: For some partition fS i g of the circle into sets consisting of …nite unions of arcs, we have (*), where denotes arc length. In this paper we show how cycle decomposition techniques originally used (Alpern, 1983) to establish Cohen's conjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, P N > 0 for some N ) stochastic matrix P can be represented (in the sense of * but with S i merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue probability space. Representations by pointwise and setwise periodic automorphisms are also established. While this paper is largely expository, all the proofs, and some of the results, are new.

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The Forward and Backward Rotational Decompositions of Markov Chains

Cited by 2 publications

References 6 publications

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

MultiTowers, conjugacies and codes: Three theorems in Ergodic theory, one variation on Rokhlin’s Lemma

Rotational (and Other) Representations of Stochastic Matrices

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