Entropy and mixing for amenable group actions.
Subshifts of Finite Type and Sofic Systems.
Generalized interval exchanges and the 2–3 conjecture
We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . A is called λ-dense if λ(A i∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that . We give necessary and sufficient conditions for this equality to hold. We show that...
Strictly ergodic, uniform positive entropy models
Locally equicontinuous dynamical systems
A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, where WAP is the class of weakly almost periodic systems and AE the class of almost equicontinuous systems....
Remarks on the tightness of cocycles
We prove a generalised tightness theorem for cocycles over an ergodic probability preserving transformation with values in Polish topological groups. We also show that subsequence tightness of cocycles over a mixing probability preserving transformation implies tightness. An example shows that this latter result may fail for cocycles over a mildly mixing probability preserving transformation.
Topological groups with Rokhlin properties
In his classical paper [Ann. of Math. 45 (1944)] P. R. Halmos shows that weak mixing is generic in the measure preserving transformations. Later, in his book, Lectures on Ergodic Theory, he gave a more streamlined proof of this fact based on a fundamental lemma due to V. A. Rokhlin. For this reason the name of Rokhlin has been attached to a variety of results, old and new, relating to the density of conjugacy classes in topological groups. In this paper we will survey some of the new developments...
Testing stationary processes for independence
Let 0 denote the class of all real valued i.i.d. processes and 1 all other ergodic real valued stationary processes. In spite of the fact that these classes are not countably tight we give a strongly consistent sequential test for distinguishing between them.
On estimating the memory for finitarily markovian processes
Coboundaries in
Forward estimation for ergodic time series
An anti-classification theorem for ergodic measure preserving transformations
Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT’s on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos–von Neumann theorem classifying ergodic MPT’s with pure point spectrum. We treat both the classical case of...
A note on prediction for discrete time series
Let be a stationary and ergodic time series taking values from a finite or countably infinite set and that is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times along which we will be able to estimate the conditional expectation from the observations in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series...
Inferring the residual waiting time for binary stationary time series
For a binary stationary time series define to be the number of consecutive ones up to the first zero encountered after time , and consider the problem of estimating the conditional distribution and conditional expectation of after one has observed the first outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state...
A note on discriminating Poisson processes from other point processes with stationary inter arrival times
We give a universal discrimination procedure for determining if a sample point drawn from an ergodic and stationary simple point process on the line with finite intensity comes from a homogeneous Poisson process with an unknown parameter. Presented with the sample on the interval the discrimination procedure , which is a function of the finite subsets of , will almost surely eventually stabilize on either POISSON or NOTPOISSON with the first alternative occurring if and only if the process is...
A versatile scheme for predicting renewal times
There are two kinds of universal schemes for estimating residual waiting times, those where the error tends to zero almost surely and those where the error tends to zero in some integral norm. Usually these schemes are different because different methods are used to prove their consistency. In this note we will give a single scheme where the average error is eventually small for all time instants, while the error itself tends to zero along a sequence of stopping times of density one.
Estimating the conditional expectations for continuous time stationary processes
One of the basic estimation problems for continuous time stationary processes , is that of estimating based on the observation of the single block when the actual distribution of the process is not known. We will give fairly optimal universal estimates of this type that correspond to the optimal results in the case of discrete time processes.
Intermittent estimation for finite alphabet finitarily Markovian processes with exponential tails
We give some estimation schemes for the conditional distribution and conditional expectation of the the next output following the observation of the first outputs of a stationary process where the random variables may take finitely many possible values. Our schemes are universal in the class of finitarily Markovian processes that have an exponential rate for the tail of the look back time distribution. In addition explicit rates are given. A necessary restriction is that the scheme proposes an...
Universal rates for estimating the residual waiting time in an intermittent way
A simple renewal process is a stochastic process taking values in where the lengths of the runs of ’s between successive zeros are independent and identically distributed. After observing one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional distribution...
Perfect filtering and double disjointness
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