Simple conditions for mixing of infinitely divisible processes

Recent advances in single-molecule experiments show that various complex systems display nonergodic behavior. In this paper, we show how to test ergodicity and ergodicity breaking in experimental data. Exploiting the so-called dynamical functional, we introduce a simple test which allows us to verify ergodic properties of a real-life process. The test can be applied to a large family of stationary infinitely divisible processes. We check the performance of the test for various simulated processes and apply it to experimental data describing the motion of mRNA molecules inside live Escherichia coli cells. We show that the data satisfy necessary conditions for mixing and ergodicity. The detailed analysis is presented in the supplementary material.

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“…The following result illustrates the strength of the dynamical functional [37,38]. The stationary ID process Y (n) is mixing if and only if…”

Section: Ergodicity Mixing and Dynamical Functionalmentioning

confidence: 99%

Anomalous diffusion: Testing ergodicity breaking in experimental data

Magdziarz

Weron

2011

Phys. Rev. E

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“…The quantity r(t) is also found in [8], where a necessary and sufficient condition for the mixing property of stationary infinitely divisible processes was proved in terms of r(t) implicitly. (See also [9].) Therefore, it is reasonable to use r(t), or equivalently, I(t) in measuring the dependence property of our stationary process {X(t)}.…”

Section: Stable Processes and Linear Fractional Stable Motionsmentioning

confidence: 99%

Long-Memory Stable Ornstein-Uhlenbeck Processes

Maejima

Yamamoto

2003

Electron. J. Probab.

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The solution of the Langevin equation driven by a Lévy process noise has been well studied, under the name of Ornstein-Uhlenbeck type process. It is a stationary Markov process. When the noise is fractional Brownian motion, the covariance of the stationary solution process has been studied by the first author with different coauthors. In the present paper, we consider the Langevin equation driven by a linear fractional stable motion noise, which is a selfsimilar process with long-range dependence but does not have finite variance, and we investigate the dependence structure of the solution process.

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“…The next result is a multivariate and random field extension of Theorem 2 of Rosinski and Zak [13] and it will help us to generalise Theorem 2.2.…”

Section: Related Results and Extensionsmentioning

confidence: 75%

“…In the present work we fill an important gap by extending the results of Maruyama [11], Rosinski and Zak [13], and Fuchs & Stelzer [6] to the multivariate random field case. First, this is crucial for applications since many of them consider a multidimensional domain composed by both spatial and temporal components (and not just temporal ones).…”

Section: Introductionmentioning

confidence: 84%

Mixing Properties of Multivariate Infinitely Divisible Random Fields

Passeggeri

Veraart

2018

J Theor Probab

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In this work we present different results concerning mixing properties of multivariate infinitely divisible (ID) stationary random fields. First, we derive some necessary and sufficient conditions for mixing of stationary ID multivariate random fields in terms of their spectral representation. Second, we prove that (linear combinations of independent) mixed moving average fields are mixing. Further, using a simple modification of the proofs of our results we are able to obtain weak mixing versions of our results. Finally, we prove the equivalence of ergodicity and weak mixing for multivariate ID stationary random fields.

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Simple conditions for mixing of infinitely divisible processes

Cited by 40 publications

References 4 publications

Anomalous diffusion: Testing ergodicity breaking in experimental data

Anomalous diffusion: Testing ergodicity breaking in experimental data

Long-Memory Stable Ornstein-Uhlenbeck Processes

Mixing Properties of Multivariate Infinitely Divisible Random Fields

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