Theđ-distance between two Markov processes cannot always be attained by a Markov joining

Ellis, Martin

doi:10.1007/bf02834757

Cited by 7 publications

(5 citation statements)

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“…For iid processes, the optimal joining problem reduces to the optimal coupling problem of their 1-dimensional marginal measures, which yields the error bound above. However, when at least one of the measures is not iid, the optimal joining need not be Markov of any order [24], and one must let k tend to infinity in order to estimate the full behavior of an optimal joining. As such, one expects to find slower rates.…”

Section: Finite-sample Error Boundmentioning

confidence: 99%

Estimation of Stationary Optimal Transport Plans

O’Connor¹,

McGoff²,

Nobel³

2021

Preprint

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We study optimal transport problems in which finite-valued quantities of interest evolve dynamically over time in a stationary fashion. Mathematically, this is a special case of the general optimal transport problem in which the distributions under study represent stationary processes and the cost depends on a finite number of time points. In this setting, we argue that one should restrict attention to stationary couplings, also known as joinings, which have close connections with long run average cost. We introduce estimators of both optimal joinings and the optimal joining cost, and we establish their consistency under mild conditions. Under stronger mixing assumptions we establish finite-sample error rates for the same estimators that extend the best known results in the iid case. Finally, we extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem.

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Section: Finite-sample Error Boundmentioning

confidence: 99%

Estimation of Stationary Optimal Transport Plans

O’Connor¹,

McGoff²,

Nobel³

2021

Preprint

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“…Moreover, we show that this explicit construction is feasible, in the sense that it can be realized by a perfect simulation algorithm which stops almost surely after a finite number of steps. on this field are Ellis (1976Ellis ( , 1978Ellis ( , 1980aEllis ( , 1980b) which consider the case of Markov chains on a finite alphabet. Ours seems to be the first constructive solution for chains of infinite order.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…6 can be seen as a generalization to the infinite volume setting of results of Kirillov et al (1989) who show that the classical coupling introduced by Holley (1974) attainsd-distance for finite volume Gibbs states. Besides Kirillov et al (1989) the only other constructive results on this field areEllis (1976Ellis ( , 1978Ellis ( , 1980aEllis ( , 1980b which consider the case of Markov chains on a finite alphabet. Ours seems to be the first constructive solution for chains of infinite order.Several challenges lay ahead.…”

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confidence: 99%

Perfect Simulation of a Coupling Achieving the $\bar{d}$ -distance Between Ordered Pairs of Binary Chains of Infinite Order

2010

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We explicitly construct a stationary coupling attaining Ornstein'sd-distance between ordered pairs of binary chains of infinite order. Our main tool is a representation of the transition probabilities of the coupled bivariate chain of infinite order as a countable mixture of Markov transition probabilities of increasing order. Under suitable conditions on the loss of memory of the chains, this representation implies that the coupled chain can be represented as a concatenation of iid sequences of bivariate finite random strings of symbols. The perfect simulation algorithm is based on the fact that we can identify the first regeneration point to the left of the origin almost surely.

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“…Since A is level, the outflow from 2 to P equals ap (2), and since the inflow into 2 is > p(l)ex2 > (a + 8)p(2), e21 > 8, so (since p(2) > 0) D is not level, whence C must be respectful. Contradiction.…”

Section: Letmentioning

confidence: 99%

“…The values of p(2) less than that for the standard process which can be attained by a Markov 9 E Y((a, ß), (y, 8)) are precisely those in Hence M* never attains ¿7 unless M already attains ¿7. (2) In the text the only method of showing that ¿7((ot, /? ), (y, 5)) =£ M ((a, ß), (y, 8)) was finding a process (a', /?')…”

Section: Letmentioning

confidence: 99%

Distances between Two-State Markov Processes Attainable by Markov Joinings

Ellis¹

1978

Transactions of the American Mathematical Society

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The function which assigns to each pair of two-state Markov processes the set of partition distances between them attainable by a Markov process on their joint atoms is computed. It is found that the infimum of these distances, the "Markov distance" between the pair, fails to satisfy the Triangle Inequality, hence fails to be a metric; thus in some cases the ¿-distance between two two-state Markov processes cannot be attained by a Markov process on their joint atoms.

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Theđ-distance between two Markov processes cannot always be attained by a Markov joining

Cited by 7 publications

References 1 publication

Estimation of Stationary Optimal Transport Plans

Estimation of Stationary Optimal Transport Plans

Perfect Simulation of a Coupling Achieving the $\bar{d}$ -distance Between Ordered Pairs of Binary Chains of Infinite Order

Distances between Two-State Markov Processes Attainable by Markov Joinings

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