Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Operator algebras associated with polymorphisms

A short survey of the problems and developments in the theory of operator algebras associated with semigroup dynamical systems is presented. The main part is an exposition of results concerning the algebras generated by multivalued transformations (polymorphisms). Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 23–27.     

Consistent Lyapunov exponent Estimation for one-dimensional dynamical systems

Summary  The author proves the consistency of a nearest neighbor estimator of the Lyapunov exponent for a general class of one-dimensional ergodic dynamical systems. The author shows that this estimator has good practical properties on a set of simulations.     

Aging of spherical spin glasses

Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for (hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging” phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures. Received: 8 July 1999 / Revised version: 2 June 2000 / Published online: 6 April 2001     

Entropy increase in dynamical systems

The rate of increase of the non-equilibrium entropy introduced by Goldstein and Penrose, defined on nonstationary probability measures for an abstract dynamical system, is quantitatively related to the Kolmogorov-Sinai entropy of the system. It is shown in particular that for ergodic systems the asymptotic rate of entropy increase coincides with the Kolmogorov-Sinai entropy. Supported in part by NSF Grant No. PHY 78-03816.     

The problem of metrizability of dynamical systems that admit normal shift

The problem of metrizability of dynamical systems that admit normal shift is formulated and solved. The explicit form of the force field for a Newtonian dynamical system _tt r=F(r, _t r) along which normal shift coincides with the metric normal shift for some conformally Euclidean metric is found.State University, Bashkir. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 85–93, October, 1994.     

Large deviations for random dynamical systems and applications to hidden Markov models

In this paper, we prove the large deviation principle (LDP) for the occupation measures of not necessarily irreducible random dynamical systems driven by Markov processes. The LDP for not necessarily irreducible dynamical systems driven by i.i.d. sequence is derived. As a further application we establish the LDP for extended hidden Markov models, filling a gap in the literature, and obtain large deviation estimations for the log-likelihood process and maximum likelihood estimator of hidden Markov models.     

Dynamical Borel-Cantelli lemmas for gibbs measures

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsA _n ⊃ X and μ-almost every pointx∈X the inclusionT ⁿ x∈A _n holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma. Partially supported by NSF grant DMS-9732728. Partially supported by NSF grant DMS-9704489.     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

On the Order of the Lebesgue Constants for Interpolation by Algebraic Polynomials from Values at Uniform Nodes of a Simplex

For interpolation processes by algebraic polynomials of degree n from values at uniform nodes of an m-simplex, where m ≥ 2, we obtain the order of growth in n of the Lebesgue constants, which coincides with that in the one-dimensional case for which Turetskii obtained an asymptotics earlier.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 814–831.Original Russian Text Copyright ©2005 by N. V. Baidakova.     

Bundles with multivalued automorphism groups

A category whose objects are principal bundles with fixed base (smooth manifold) B, structure group T ^k , and finite group of multivalued automorphisms is constructed; the morphisms are required to be equivariant with respect to . Invariants are found and used to calculate the group of equivalence classes of the category objects. Examples are given and applications to dynamical systems with gyroscopic forces are suggested.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 600–616.Original Russian Text Copyright © 2005 by A. V. Ryzhkova, E. I. Yakovlev.This revised version was published online in April 2005 with a corrected issue number.     

Optimal unbiased estimators for finite families of distributions

Necessary and sufficient conditions for the existence of a nontrivial optimal algebra are established on the basis of a statistical structure with finitely many measures. A method is proposed for constructing an optimal unbiased estimator of a given function of a parameter, if it exists.Translated from Statisticheskie Metody, pp. 156–163, 1980.     

On the existence of generalized gibbs measures for the one-dimensional <Emphasis Type="Italic">p</Emphasis>-adic countable state Potts model

We consider the one-dimensional countable state p-adic Potts model. A construction of generalized p-adic Gibbs measures depending on weights λ is given, and an investigation of such measures is reduced to the examination of a p-adic dynamical system. This dynamical system has a form of series of rational functions. Studying such a dynamical system, under some condition concerning weights, we prove the existence of generalized p-adic Gibbs measures. Note that the condition found does not depend on the values of the prime p, and therefore an analogous fact is not true when the number of states is finite. It is also shown that under the condition there may occur a phase transition.     

A simple characterization of the set ofμ-entropy pairs and applications

We present simple characterizations of the setsE _μ andE _X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE _X . Finally an example of a minimal point distal dynamical system is constructed for whichE _X ∩(X ₀×X ₀)≠ , whereX ₀ is the denseG _δ subset of distal points inX.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

Test of association between multivariate stable vectors

Based on observations of d-dimensional random vectors in the domain of attraction of a stable distribution with (multi-)index α = (α₁, …, α_d), an estimator for the dependence function of the α_i-stable variables is constructed. The estimator utilizes the α-tail-estimator and an estimator of the spectral measure of the α-stable law. This estimator gives rise to a test of association of the stable components and various quantitative measures of association.     

Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes

We consider the problem of estimating the covariance of two diffusion-type processes when they are observed only at discrete times in a nonsynchronous manner. In our previous work in 2003, we proposed a new estimator which is free of any ‘synchronization’ processing of the original data and showed that it is consistent for the true covariance of the processes as the observation interval shrinks to zero; Hayashi and Yoshida (Bernoulli, 11, 359–379, 2005). This paper is its sequel. Specifically, it establishes asymptotic normality of the estimator in a general nonsynchronous sampling scheme.     

Multidimensional dynamical systems that admit normal shift

Dynamical systems in R _n of the formR ⁿ of the form that admit normal shift of submanifolds of codimension 1 are considered. The concept of weak normality for such systems is defined, and defining partial differential equations for the force function of dynamical systems with the properties of weak and complete normality are derived.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 264–269, August, 1994.