Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness

Hubert Hennion, Loic Herve9783540424154, 3-540-42415-6

This book shows how techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel, may be used to obtain limit theorems for Markov chains or to describe stochastic properties of dynamical systems.A general framework for this method is given and then applied to treat several specific cases. An essential element of this work is the description of the peripheral spectra of a quasi-compact Markov kernel and of its Fourier-Laplace perturbations. This is first done in the ergodic but non-mixing case. This work is extended by the second author to the non-ergodic case.The only prerequisites for this book are a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis.

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Table of contents :
Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness……Page 1
PREFACE……Page 3
TALBLE OF CONTENTS……Page 4
I.2. STOCHASTIC PROPERTIES OF DYNAMICAL SYSTEMS……Page 8
I.3. HISTORICAL BACKGROUND TO THE METHOD……Page 9
I.4. PURPOSE OF THE PAPER……Page 10
II.2. CONDITIONS H[m] AND D NOTATIONS N……Page 15
II.3. STATEMENT OF VARIOUS CENTRAL LIMIT THEOREMS……Page 18
III.2. A PERTURBATION THEOREM……Page 25
IV.2. CENTRAL LIMIT THEOREM: INTERMEDIATE RESULT……Page 35
V.2. PERIPHERAL EIGENVALUES OF Q(t) FOR SMALL Itl……Page 42
VI.3. C. L. T. WITH A RATE OF CONVERGENCE (Th. B)……Page 48
VI.4. LOCAL CENTRAL LIMIT THEOREM (Theorem C)……Page 50
VII.2. PROOF OF THEOREM VII.2……Page 54
VIII.2. PROPERTIES OF LAPLACE KERNELS, FUNCTION c……Page 60
VIII.3. LOGARITHMIC ESTIMATE: THEOREM E-(i)-(ii)……Page 62
VIII.4. PROBABILITY OF A LARGE DEVIATION: Th. E-(iii)……Page 64
VIII.5. ADDITIONAL STATEMENTS……Page 68
X.2. INVARIANT DISTRIBUTIONS, QUASI-COMPACTNESS……Page 76
X.3. LAPLACE KERNELS……Page 82
X.4. PRODUCTS OF INVERTIBLE RANDOM MATRICES……Page 87
X.5. PRODUCTS OF POSITIVE RANDOM MATRICES……Page 90
X.6. AUTOREGRESIVE PROCESSES……Page 91
XI.2. r–INVARIANT DISTRIBUTION, RELATIVIZED KERNEL……Page 96
XI.3. PROOFS OF THE LIMIT THEOREMS……Page 98
XII.2 – SUBSHIFTS AND TRANSFER OPERATORS……Page 105
XIII.2. PROOF OF LEMMA IV-5……Page 113
XIII.3. LARGE DEVIATIONS LEMMA……Page 114
XIV.1. A CONDITION FOR QUASI-COMPACTNESS……Page 117
XIV.2. PROOF OF THE PERTURBATION THEOREM (Th. 111-8)……Page 124
1. Hypotheses and notations……Page 128
2. Limit theorems for Markov chains……Page 129
2.1. Limit theorems for the sequence ( theta(X_k))_k……Page 130
2.2. Some elements of the proofs of limit theorems……Page 133
3.1. Case B=L(E) (Lipschitz functions)……Page 136
3.2. Sufficient conditions for (K2)(ii)……Page 137
3.3. Investigation of the conditions (K3) and D……Page 140
4. Applications to dynamical systems……Page 141
5.1. Q-invariant subsets and induced operators……Page 144
5.3. Proof of Theorem 2.3……Page 145
REFERENCES……Page 153
INDEXES……Page 157