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Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions
 PROBABILITY SURVEYS
, 2005
"... This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions. That paper appeared in 1986 in a book containing survey papers on various types of dependence conditions and the limit theory under them. The survey here will include pa ..."
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Cited by 46 (0 self)
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This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions. That paper appeared in 1986 in a book containing survey papers on various types of dependence conditions and the limit theory under them. The survey here will include part (but not all) of the material in [18], and will also describe some relevant material that was not in that paper, especially some new discoveries and developments that have occurred since that paper was published. (Much of the new material described here involves “interlaced ” strong mixing conditions, in which the index sets are not restricted to “past ” and “future.”) At various places in this survey, open problems will be posed. There is a large literature on basic properties of strong mixing conditions. A survey such as this cannot do full justice to it. Here are a few references on important topics not covered in this survey. For the approximation of mixing sequences by martingale differences, see e.g. the book by Hall and Heyde [80]. For the direct approximation of mixing random variables by independent ones,
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 15 (7 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
Nonparametric Entropy Estimation for Stationary Processes and Random Fields, with Applications to English Text
, 1998
"... We discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a Doeblintype mixing condition. The estimators are Ces`aro averages of longest matchlengths, and their consistency follows from a generalized ergodic theorem ..."
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Cited by 15 (5 self)
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We discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a Doeblintype mixing condition. The estimators are Ces`aro averages of longest matchlengths, and their consistency follows from a generalized ergodic theorem due to Maker. We provide examples of their performance on English text, and we generalize our results to countable alphabet processes and to random fields.
Uniform ergodic theorems for dynamical systems under VC entropy conditions
 Proc. Probab. Banach Spaces IX (Sandbjerg
, 1993
"... Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann’s mean and Birkhoff’s pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of b ..."
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Cited by 3 (2 self)
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Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann’s mean and Birkhoff’s pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally
An approach to the existence of unique invariant probabilities for Markov processes. Limit theorems in probability and statistics
 János Bolyai Math. Soc., I (Balatonlelle
, 1999
"... A notion of localized splitting is introduced as a further extension of the splitting notions for iterated monotone maps introduced earlier by Dubins and Freedman (1966) and more generally by Bhattacharya and Majumdar (1999). We will see that under quite general conditions, localized splitting theor ..."
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Cited by 3 (0 self)
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A notion of localized splitting is introduced as a further extension of the splitting notions for iterated monotone maps introduced earlier by Dubins and Freedman (1966) and more generally by Bhattacharya and Majumdar (1999). We will see that under quite general conditions, localized splitting theory is a natural extension of the Doeblin (1937) minorization theory, Harris (1956) recurrence theory, splitting theory of Nummelin (1978) and regeneration theory of Athreya and Ney (1978), under which we can prove the existence of a unique invariant probability. By way of introduction we also provide a natural coupling proof of the ergodic problem for Markov processes on general state spaces under Doeblin’s minorization condition which seems to have heretofore gone unnoticed. The paper is concluded with some new applications of splitting theory to random iterated quadratic maps. 1
Small sets and Markov transition densities
, 2002
"... The theory of general statespace Markov chains can be strongly related to the case of discrete statespace by use of the notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on statespaces with countably generated σalgeb ..."
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Cited by 2 (2 self)
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The theory of general statespace Markov chains can be strongly related to the case of discrete statespace by use of the notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on statespaces with countably generated σalgebras, though the minorization provided by the theory concerns small sets of order n and nstep transition kernels for some unspecified n. Partly motivated by the growing importance of small sets for Markov chain Monte Carlo and Coupling from the Past, we show that in general there need be no small sets of order n = 1 even if the kernel is assumed to have a density function (though of course one can take n = 1 if the kernel density is continuous). However n = 2 will suffice for kernels with densities (integral kernels), and in fact small sets of order 2 abound in the technical sense that the 2step kernel density can be expressed as a countable sum of nonnegative separable summands based on small sets. This can be exploited to produce a representation using a latent discrete Markov chain; indeed one might say, inside every Markov chain with measurable transition density there is a discrete statespace Markov chain struggling to escape. We conclude by discussing complements to these results, including their relevance to Harrisrecurrent Markov chains and we relate the counterexample to Turán problems for bipartite graphs.
Brownian Motion with Restoring Drift: Microcanonical Ensemble and the Thermodynamic Limit
"... We take up the old problem of microcanonical conditioning in the context of diusion. Starting with a potential F : R ! R, the Schrodinger operator G 0 = (1=2)4 F with ground state is carried by a conjugation into the diusion generator G = (1=2)4+ (r = ) r with invariant density . The latter ..."
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We take up the old problem of microcanonical conditioning in the context of diusion. Starting with a potential F : R ! R, the Schrodinger operator G 0 = (1=2)4 F with ground state is carried by a conjugation into the diusion generator G = (1=2)4+ (r = ) r with invariant density . The latter motion t ! X(t) is made microcanonical by rst conditioning the path to be periodic, X(0) = X(L), and then further conditioning on the empirical meansquare or \particle number" (1=L) dt ' D. The thermodynamics are then studied by taking L " 1 while D remains xed. The problem in this form owes its inception to McKeanVaninsky [8] who obtained the following result. For F (x)=jxj " 1 with jxj " 1, they showed the same type of diusion appears in the thermodynamic limit, but with drift arising from the shifted potential F + cjxj , c being such that the limiting meansquare equals D. Their method of proof predicts the same outcome for F (x)=jxj # 0, so long as D is smaller than the canonical meansquare D 0 = (x)dx, while if D > D 0 , the matter was unresolved. The purpose of this note is to show a type of phase transition takes place in this case: the conditioning is overcome in the limit and one sees the original (stationary) diusion on the line. The proof employs an entropy inequality due to Csiszar [1]. 1.
Eds., vol. 2 pp. 27–48.
"... with, for example, neural nets. The same numerical search methods are applicable for both of these model structures. REFERENCES [1] L. Breiman, “Hinging hyperplanes for regression, classification and function approximation, ” IEEE Trans. Inform. Theory, vol. 39, pp. ..."
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with, for example, neural nets. The same numerical search methods are applicable for both of these model structures. REFERENCES [1] L. Breiman, “Hinging hyperplanes for regression, classification and function approximation, ” IEEE Trans. Inform. Theory, vol. 39, pp.
Correspondence DoeblinHostinsk´y 1 Ten letters from Wolfgang Doeblin
"... Observation: We present the 10 letters sent by Doeblin to Hostinsk´y between 1936 and 1938, ..."
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Observation: We present the 10 letters sent by Doeblin to Hostinsk´y between 1936 and 1938,