On two–block–factor sequences and one–dependence

Matúš, František

doi:10.1090/s0002-9939-96-03094-8

Cited by 7 publications

(3 citation statements)

References 8 publications

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“…8 In fact, a Markov process can be q-dependent. Lévy (1949), Rosenblatt and Slepian (1962), Aaronson, Gilat, and Keane (1992), and Matús (1996and Matús ( , 1998 provide examples of a q-dependent Markov process. Ibragimov (2007) provides the conditions that a Markov process is a q-dependent process.…”

Section: Asymptotic Distributionmentioning

confidence: 99%

Testing for the Markov Property in Time Series

Chen¹,

Hong²

2011

Econom. Theory

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The Markov property is a fundamental property in time series analysis and is often assumed in economic and financial modeling. We develop a new test for the Markov property using the conditional characteristic function embedded in a frequency domain approach, which checks the implication of the Markov property in every conditional moment (if it exists) and over many lags. The proposed test is applicable to both univariate and multivariate time series with discrete or continuous distributions. Simulation studies show that with the use of a smoothed nonparametric transition density-based bootstrap procedure, the proposed test has reasonable sizes and all-around power against several popular non-Markov alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the Markov property.

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Section: Asymptotic Distributionmentioning

confidence: 99%

Testing for the Markov Property in Time Series

Chen¹,

Hong²

2011

Econom. Theory

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“…While finitely dependent processes enjoyed significant attention in the intervening years [1, 2, 16-18, 23, 35, 36, 53], it was only in 1993 that Ibragimov and Linnik's question was resolved in the affirmative by Burton, Goulet, and Meester [14]. Many subsequent works [11,13,19,37,43,46,47] explored the properties of such processes, but the question remained: are there 'natural' stationary finitely dependent processes that are not block-factors?…”

Section: Introductionmentioning

confidence: 99%

Mallows permutations and finite dependence

Holroyd¹,

Hutchcroft²,

Levy³

2020

Ann. Probab.

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We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.

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“…This question and the surrounding issues have been taken up by a number of authors [1, 7-10, 14, 16, 18, 24, 26, 31, 34, 35, 40], and various further examples have been constructed. Highlights include an explicit 1-dependent (5state) Markov chain that is not a 2-block factor [1], a (hidden-Markov) 1-dependent process that is not an r-block-factor for any r (Burton, Goulet and Meester, 1993 [8]), and a "perturbable" example showing that 2-block-factors are not dense in the set of 1-dependent Markov chains [34].…”

Section: Introductionmentioning

confidence: 99%

Finitely Dependent Coloring

Holroyd

Liggett

2016

Forum of Mathematics, Pi

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We prove that proper coloring distinguishes between block factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently wellseparated locations are independent; it is a block factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and asked whether a k-dependent q-coloring exists for any k and q. We give a complete answer by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovász local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving d dimensions and shifts of finite type; in fact, any nondegenerate shift of finite type also distinguishes between block factors and finitely dependent processes.

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On two–block–factor sequences and one–dependence

Cited by 7 publications

References 8 publications

Testing for the Markov Property in Time Series

Testing for the Markov Property in Time Series

Mallows permutations and finite dependence

Finitely Dependent Coloring

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