The Gibbs Conditioning Principle for Markov Chains

Meda, Ana; Ney, Peter

doi:10.1007/978-1-4612-2168-5_21

Cited by 4 publications

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“…There exist several refinements of this result describing rigorously the precise asymptotic strategy by which the random variables realize the large deviation event {L n ∈ A}. See the book of Dembo and Zeitouni [6] for more on the classical Gibbs conditioning principle, [5,7,14] for refinements, and [8,11,12] for further examples of conditioning principles.…”

Section: Introductionmentioning

confidence: 99%

Galton–Watson Trees with Vanishing Martingale Limit

et al. 2014

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Abstract:We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than ε, agrees up to generation K with a regular µ-ary tree, where µ is the essential minimum of the offspring distribution and the random variable K is strongly concentrated near an explicit deterministic function growing like a multiple of log(1/ε). More precisely, we show that if µ ≥ 2 then with high probability as ε ↓ 0, K takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular µ-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.Mathematics Subject Classification (2010): 60J80 (Primary) 60F10, 60K37.

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Section: Introductionmentioning

confidence: 99%

Galton–Watson Trees with Vanishing Martingale Limit

et al. 2014

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“…Note that bounded convergence is convergence in the τ -topology. Note also that this convergence is the same if defined for functions f : S → R (instead of R d , as we did in [16]). …”

Section: Conditional Weak Laws In Banach Spaces 2601mentioning

confidence: 52%

“…in R d see, for example, [19], [12] and [13], [6]. For general sequences of the Gartner-Ellis type and for the Markov case in R d see [14], [15], [16]; for the i.i.d. case in Banach spaces see [4].…”

Section: Introductionmentioning

confidence: 99%

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Conditional weak laws in Banach spaces

Meda

2002

Proc. Amer. Math. Soc.

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Thermodynamical Aspects of Classical Lattice Systems

Pfister

2002

In and Out of Equilibrium

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In these lectures I consider the mathematical formalism of thermodynamics and statistical mechanics for lattice systems and show its tight connection with large deviations theory. The models I have in mind are the lattice models with compact state spaces, although the theory is developed in greater generality. The central notion studied here is the notion of an asymptotically decoupled p r obability measure. The main theme of these lectures is the thermodynamical like aspects of the law of large numbers. The main feature of the asymptotically decoupled probability measures is that the existence of important q u a n tities, like the speci c information gain, which h a ve a thermodynamical interpretation in statistical mechanics, can be established using the standard and simple arguments developed for proving the existence of thermodynamical limits (see e.g. YL] 1 ). A large part of these lectures is original. However, the principal results and methods have b e e n d e v eloped much earlier in a less general setting 2 . My presentation is strongly in uenced by m y longstanding collaboration with John Lewis and Wayne Sullivan in Dublin. It is based principally on LP] and LPS3]. The notion of asymptotically decoupled measures 3 has been introduced in lectures delivered in Grenoble in Spring 1999 on the same subject.Section 2 serves as an introductory section. I consider a special model and present the main features of statistical mechanics of lattice systems. The interplay b e t ween mathematical physics and large deviations theory is displayed. No proof is given, but almost all results presented in that section are proved with complete proofs in much greater generality in sections 3 to 5. Section 2 is independent of the rest of the lectures.In section 3 asymptotically decoupled measures are de ned and examples are given. Large deviations properties of such measures are expressed through two functionals: the speci c information gain h( j ), which is de ned on the space of translation invariant probability measures M + 1 , a n d p( j ) (called the pressure), which is de ned on the dual space of M + 1 , the space of the quasilocal functions F qloc . These functionals are conjugate to each other. Convexity p l a ys an important role in the thermodynamical formalism. The solutions of the equation h( j ) = 0 and the related notion of asymptotically I-null sequences, extending the notion of asymptotically quasi-independence of Csisz ar, are then studied. In subsection 3.4 a large deviations principle for the empirical measures in the multidimensional case is established.In section 4 the space L of bounded total oscillation functions is studied. L is a dense subset of the quasilocal functions. Nice properties are true on L. In particular 1 The importance of the thermodynamical limit in statistical mechanics was recognized in the late thirties. See in particular the very interesting introduction of KU].2 I did not write a review paper. The bibliography i s v ery incomplete, and the historical remarks very rare. More inform...

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The Gibbs Conditioning Principle for Markov Chains

Cited by 4 publications

References 20 publications

Galton–Watson Trees with Vanishing Martingale Limit

Galton–Watson Trees with Vanishing Martingale Limit

Conditional weak laws in Banach spaces

Thermodynamical Aspects of Classical Lattice Systems

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