Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems an...

Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak...

The present work is an endeavour to determine analytically features of the stationary measure of a non-integrable zero-range process, and to investigate the possible existence of phase transitions for such a nonequilibrium model. The rates defining the model do not satisfy the constraints necessary for the stationary measure to be a product measure. Even in the absence of a drive, detailed bala...

The northeast model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: Whenever a site’s southerly and westerly nearest neighbors have spin 1, it may reset its own spin by tossing a p−coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at pc = 1 − βc, where βc is the critical parameter...

We study attractive particle systems with stationary product measures. We utilize the property of attractivity and its link to coupling to build a growth process that samples from the stationary measure of the zero-range process, on fixed and finite lattices, with computation times scaling linearly with the number of particles N . The zero-range process with constant jump-rates and a single def...

We recall that a probability measure ν on X is said to be μ-stationary if one has μ ∗ ν = ν. It is then said to be μ-ergodic if it is extremal among μ-stationary probability measures. We will say that a probability measure ν on X is homogeneous if it is supported by a closed orbit F of its stabilizer Gν := {g ∈ G | g∗ν = ν}. Such a probability is a finite average of probability measures which a...

We present data-dependent learning bounds for the general scenario of non-stationary nonmixing stochastic processes. Our learning guarantees are expressed in terms of a datadependent measure of sequential complexity and a discrepancy measure that can be estimated from data under some mild assumptions. We also also provide novel analysis of stable time series forecasting algorithm using this new...

Investigating the entropy distance between the Wiener measure,Wt0,τ , and stationary Gaussian measures, Qt0,τ on the space of continuous functions C[t0 − τ, t0 + τ ], we show that in some cases this distance can essentially be computed. This is done by explicitly computing a related quantity which in effect is a valid approximation of the entropy distance, provided it is sufficiently small; thi...

We consider the inviscid limit of the stochastic damped 2D NavierStokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier-Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The...

We present an improvement of an estimator of causality in financial time series via transfer entropy, which includes the side information that may affect the cause-effect relation in the system, i.e. a conditional information-transfer based causality. We show that for weakly stationary time series the conditional transfer entropy measure is nonnegative and bounded below by the Geweke’s measure ...