Ergodic Case Research Articles

Tail-invariant measures for some suspension semiflows

We consider suspension semiflows over abelian extensions of one-sided mixing subshifts of finite type. Although these are not uniquely ergodic, we identify (in the ergodic case) all tail-invariant, locally finite measures which are quasiinvariant for the semiflow.

Quenched, annealed and functional large deviations for one-dimensional random walk in random environment

Suppose that the integers are assigned random variables {ω i} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X n} (called a RWRE) which, when at i, moves one step to the right with probability ω i, and one step to the left with probability 1 − ω i. When the {ω i} sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for X n/n, conditional upon the environment, with deterministic rate function.We consider in this paper large deviations, both conditioned on the environment (quenched) and averaged on the environment (annealed), for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle.

Large Deviations for Expanding Transformations with Additive White Noise

Large-deviations estimates for the autocorrelations of order kof the random process Z n=φ(X n)+ξ n, n≥0, are obtained. The processes (X n) n≥0and (ξ n) n≥0are independent, ξ n, n≥0, are i.i.d. bounded random variables, X n=T n(X 0), n∈ $$\mathbb{N}$$ , T: M→Mis expanding leaving invariant a Gibbs measure on a compact set M, and φ: M→ $$\mathbb{R}$$ is a continuous function. A possible application of this result is the case where Mis the unit circle and the Gibbs measure is the one absolutely continuous with respect to the Lebesgue measure on the circle. The case when Tis a uniquely ergodic map was studied in Carmona et al.(1998). In the present paper Tis an expanding map. However, it is possible to derive large-deviations properties for the autocorrelations samples (1/n) ∑ n−1 j=0 Z j Z j+k . But the deviation function is quite different from the uniquely ergodic case because it is necessary to take into account the entropy of invariant measures for Tas an important information. The method employed here is a combination of the variational principle of the thermodynamic formalism with Donsker and Varadhan's large-deviations approach.

Point-Stationarity in d Dimensions and Palm Theory

This paper extends to d> 1 dimensions the concept of point-stationarity, which formalizes the intuitive idea of a point process for which the behaviour relative to a given point of the process is independent of the point selected as origin. After defining point-stationarity, this concept is characterized in several ways and the characterizations then used to extend to d dimensions a particular approach to Palm theory, producing two dualities between stationary and point-stationary processes with quite different interpretations. The dualities coincide in the ergodic case.

Linear aggregation in cointegrated systems

In this paper, the problem of aggregation of integrated time series variables is contrasted with the case of aggregation of stationary variables. In the integrated case, it is seen that a cointegration condition is sufficient for aggregation bias to vanish asymptotically; no analogous condition is available in the stationary and ergodic case. Thus tests for aggregation bias, originally designed for stationary data, can be misleading when applied to integrated data. This may account for overrejection of the null hypothesis of no aggregation bias, when the null is true.

Statistical Properties of Matrix Elements in a Hamilton System Between Integrability and Chaos

This paper concerns the statistical properties of matrix elements of Hermitian operators in an eigenbasis of a typical bounded Hamiltonian, which can be classically either ergodic, or integrable, or somewhere in between (mixed dynamics). The various types of statistics are addressed: the (one-)matrix element probability distribution, the correlations among different matrix elements and the so-called cross-correlations of matrix elements belonging to different operators. A sort of n-point correlation function among n matrix elements is defined and calculated (which can reduce to the generalized Feingold-Peres formula for the mean square magnitude of matrix elements in the special case of n = 2) and it is shown that in the semiclassical limit it tends to a microcanonical average over analog expressions involving classical observables. The semiclassical O(ħ) corrections yield interesting purely classical relations expressing microcanonical averages involving different time Poisson brackets as energy derivatives of microcanonical averages involving only ordinary products. In the classically ergodic case random matrix theory can be applied for modeling of all types of statistics, although the newly defined cross-statistics seem to be behaving nonuniversally at first sight.

Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator. Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

Adaptive tests for stochastic processes in the ergodic case

A hypothesis testing problem for two processes is considered. A test which is asymptotically efficient for specified pairs of null hypotheses and alternatives is called adaptive. A necessary and sufficient adaptation condition in the ergodic case is obtained. Several examples with explicitly constructed adaptive tests are studied.

Fractional differentiation in the self-affine case I – Random functions

The invariance structure of self-affine functions and measures leads to the concept of fractional Cesáro derivatives and densities, respectively. In the present paper the case of random functions from R p into R q is considered. It is shown that the corresponding derivatives exist a.s. and equal a constant in the ergodic case. Part II will deal with the class of self-similar extremal processes and certain extensions. In Part III the fractional density of the Cantor measure will be evaluated, and arbitrary self-similar random measures will be treated in Part IV. There exists a deeper connection to fractional differentiation in the theory of function spaces which will be established elsewhere.

Exact formula for sensitivity analysis of Markov chains

Given a family of Markov chains whose transition matrices depend on a parameter vector, we given an exact formula for the gradient of the equilibrium distribution with respect to that parameter even in the case of multiple ergodic classes and transient states. This formula generalizes previous results in the ergodic case.

ERGODIC DECOMPOSITION OF FLOWS ON HOMOGENEOUS SPACES OF FINITE VOLUME

The reduction of the theory of flows on homogeneous spaces of finite volume to the ergodic case is studied in full scope. Bibliography: 20 titles.

Expansion of the global error for numerical schemes solving stochastic differential equations

Given the solution (Xt ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(Xt ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (Xt ) by simulating a simple t,rajectory. For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E. Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy

Random perturbations of transformations of an interval

Let μe be invariant measures of the Markov chainsx n F which are small random perturbations of an endomorphismf of the interval [0,1] satisfying the conditions of Misiurewicz [6]. It is shown here that in the ergodic case μe converges as e→0 to the smoothf-invariant measure which exists according to [6]. This result exhibits the first example of stability with respect to random perturbations while stability with respect to deterministic perturbations does not take place.

Maximum likelihood estimation for doubly stochastic poisson processes with partial observations

This paper deals with a nonlinear filtering approach to the problem of intensity parameter estimation of a doubly stochastic Poisson process, driven by an unob‐ servable Markov process. We present a method for the evaluation of the conditional likelihood ratio, given the observations of one path of the point process. By the methodsof Ibragimov, Khasminskii and Kutoyants we investigate the asymptotic properties of the corresponding maximum likelihood estimator. In the ergodic case consistency, asymptotic normality, convergence of the moments and asymptotic efficiency are established.

A return time invariant for finitary isomorphisms

AbstractPoincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

An application of number theory to ergodic theory and the construction of uniquely ergodic models

Using a combinatorial result of N. Hindman one can extend Jewett’s method for proving that a weakly mixing measure preserving transformation has a uniquely ergodic model to the general ergodic case. We sketch a proof of this reviewing the main steps in Jewett’s argument.

Martingales, the smoothing index and ergodic theory

A classical theorem of Meyer Jerison which shows that the convergence in the pointwise ergodic theorem is equivalent to the convergence of an associated martingale is expanded to a conditional setting. An equiconvergence theorem of the type established for martingales by N.F.G. Martin and E. Boylan is established in the ergodic case for an ergodic, non-invertible, measure-preserving transformation.

Integrals of Motion

The properties of conservative dynamical systems of two or more degrees of freedom are reviewed. The transition from integrable to ergodic systems is described. Nonintegrability is due to the interaction of two, or more, resonances. Then one sees, on a surface of section, infinite types of islands of various orders, while the asymptotic curves from unstable invariant points intersect each other along homoclinic and heteroclinic points producing an apparent ‘dissolution’ of the invariant curves. A threshold energy is defined separating near integrable systems from near ergodic ones. The possibility of real ergodicity for large enough energies is discussed. In the case of many degrees of freedom we also distinguish between integrable, ergodic, and intermediate cases. Among the latter are systems of particles interacting with Lennard-Jones interparticle potential. A threshold energy was derived, which is proportional to the number of particles. Finally some recent results about the general three-body problem are described. One can extend the families of periodic orbits of the restricted problem to the general three-body problem. Many of these orbits are stable. An empirical study of orbits near the stable periodic orbits indicates the existence of 2 integrals of motion besides the energy.

Validity of approximate methods in linear response theory for an interacting spin-phonon system

In order to gain insight into their degree of validity, certain commonly used approximate methods for finding dynamic susceptibilities are compared with the exact Kubo susceptibility for a linearly interacting spin-phonon system. Approximate methods investigated include: truncation or decoupling of equations of motion for the commutator and anticommutator Green's functions, and a memory-function method suggested by G\otze and W\olfle. It is shown that approximate methods, while adequate for weak dipole-phonon coupling, can lead to conspicuous deviations from the exact susceptibility in certain frequency ranges. Results are also sensitive to the shape of the phonon spectrum. Ergodic and nonergodic cases are discussed.

The recognition problem with time constraints

A RECOGNITION system with a fixed recognition time and a fixed maximum permissible waiting time is considered. The extremum of the probability of the exact classification of such a system as a function of the recognition time is investigated. The ergodic case is investigated. For the same case a study is made of the conditions of the closeness of the probability of exact classification of the recognition system with Gaussian time constraints to the probability of exact classification of a recognition system with fixed constraints.