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E-raamat: Nonconventional Limit Theorems And Random Dynamics

(Hebrew Univ Of Jerusalem, Israel), (Hebrew Univ Of Jerusalem, Israel)
  • Formaat: 300 pages
  • Ilmumisaeg: 09-Apr-2018
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813235021
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Nonconventional Limit Theorems And Random DynamicsSuurem pilt
  • Formaat: 300 pages
  • Ilmumisaeg: 09-Apr-2018
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813235021
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The book is devoted to limit theorems for nonconventional sums and arrays. Asymptotic behavior of such sums were first studied in ergodic theory but recently it turned out that main limit theorems of probability theory, such as central, local and Poisson limit theorems can also be obtained for such expressions. In order to obtain sufficiently general local limit theorem, we develop also thermodynamic formalism type results for random complex operators, which is one of the novelties of the book.
Preface v
Nonconventional Limit Theorems 1(2)
1 Stein's method in the nonconventional setup
3(62)
1.1 Introduction: local (strong) dependence structure
3(2)
1.2 Stein's method for normal aproximation
5(6)
1.2.1 A short introduction
5(1)
1.2.2 Normal approximation for graphical indexation
6(2)
1.2.3 Weak local dependence coefficients
8(2)
1.2.4 Relations with more familiar mixing coefficients
10(1)
1.3 Nonconventional CLT with convergence rates
11(21)
1.3.1 Assumptions and main results
11(2)
1.3.2 Asymptotic variance
13(2)
1.3.3 CLT with convergence rate
15(1)
1.3.4 The associated strong dependency graphs
16(2)
1.3.5 Expectation estimates
18(6)
1.3.6 Proof of Theorem 1.3.7
24(2)
1.3.7 Back to graphical indexation
26(6)
1.4 General Stein's estimates: proofs
32(11)
1.4.1 Proof of Theorem 1.2.1
32(4)
1.4.2 Proof of Theorem 1.2.2
36(7)
1.5 Stein's method for diffusion approximations
43(8)
1.5.1 A functional CLT via Stein's method
43(5)
1.5.2 Finite dimensional convergence rate
48(3)
1.6 A nonconventional functional CLT
51(5)
1.7 Extensions to nonlinear indexes
56(9)
1.7.1 Preliminaries
56(2)
1.7.2 Another example with nonlinear indexes
58(7)
2 Local limit theorem
65(72)
2.1 Introduction
65(2)
2.2 Local central limit theorem via Fourier analysis
67(4)
2.3 Nonconventional LLT for Markov chains by reduction to random dynamics
71(3)
2.4 Markov chains with densities
74(7)
2.4.1 Basic assumptions and CLT
74(2)
2.4.2 Characteristic functions estimates
76(5)
2.5 Markov chains related to dynamical systems
81(9)
2.5.1 Locally distance expanding maps and transfer operators
81(1)
2.5.2 Inverse branches, the pairing property and periodic points
82(2)
2.5.3 Thermodynamic formalism constructions and the associated Markov chains
84(2)
2.5.4 Relations between dynamical systems and Markov chains
86(2)
2.5.5 Mixing and approximation assumptions
88(1)
2.5.6 Asymptotic variance and the CLT
89(1)
2.6 Statement of the local limit theorem
90(2)
2.7 The associated random transfer operators
92(7)
2.7.1 Random complex RPF theorem
93(2)
2.7.2 Distortion properties
95(2)
2.7.3 Reduction to random dynamics
97(2)
2.8 Decay of characteristic functions for small t's
99(14)
2.8.1 The random pressure function
99(2)
2.8.2 The derivatives of the pressure
101(2)
2.8.3 Pressure near 0
103(1)
2.8.4 Norms estimates: employing the pressure
104(7)
2.8.5 Remarks
111(2)
2.9 Decay of characteristic functions for large t's
113(13)
2.9.1 Basic estimates and strategy of the proof
114(1)
2.9.2 Probabilities of large number of visits to open sets
115(3)
2.9.3 Segments of periodic orbits
118(3)
2.9.4 Parametric continuity of transfer operators
121(2)
2.9.5 Quasi compactness of Rit, lattice and non-lattice cases
123(1)
2.9.6 Norms estimates
124(2)
2.10 The periodic point approach to quenched and annealed dynamics
126(2)
2.11 Extensions to dynamical systems
128(9)
2.11.1 Subshifts of finite type
128(1)
2.11.2 The strategy of the proof
129(1)
2.11.3 Local limit theorem: one sided case
130(1)
2.11.4 Two sided case
131(1)
2.11.5 Reduction to one sided case
132(5)
3 Nonconventional arrays
137(34)
3.1 Introduction
137(2)
3.2 Strong law of large numbers for nonconventional arrays
139(10)
3.2.1 Setup and the main result
139(2)
3.2.2 Auxiliary lemmas
141(3)
3.2.3 Ordering and decompositions
144(2)
3.2.4 Proof of Theorem 3.2.2
146(3)
3.3 Central limit theorem
149(10)
3.3.1 L.i.D. case
149(3)
3.3.2 Convergence of covariances
152(4)
3.3.3 The number of solutions
156(1)
3.3.4 Martingale approximation
157(2)
3.4 Poisson limit theorems for nonconventional arrays
159(12)
3.4.1 Preliminaries and main results
159(3)
3.4.2 Stationary sequences
162(4)
3.4.3 Poisson limits for subshifts
166(5)
Thermodynamic Formalism for Random Complex Operators and applications
171(98)
4 Random complex Ruelle-Perron-Frobenius theorem via cones contractions
173(24)
4.1 Preliminaries
173(2)
4.2 Main results
175(2)
4.3 Block partitions and RPF triplets
177(7)
4.3.1 Reverse block partitions
177(6)
4.3.3 RPF triplets
183(1)
4.4 Exponential convergences
184(5)
4.4.1 Taylor reminders and important bounds
185(1)
4.4.2 Exponential convergences
186(2)
4.4.3 Additional types of exponential convergences
188(1)
4.5 Uniqueness of RPF triplets
189(2)
4.6 The largest characteristic exponents
191(6)
4.6.1 Analyticity of the largest characteristic exponent around 0
192(1)
4.6.2 The pressure function
193(1)
4.6.3 Proof of Theorem 4.6.3
194(3)
5 Application to random locally distance expanding covering maps
197(34)
5.1 Random locally expanding covering maps
197(4)
5.2 Transfer operators
201(1)
5.3 Real and complex cones
202(1)
5.4 RPF triplets
203(2)
5.5 Properties of cones: proof of Theorem 5.3.1
205(5)
5.6 Properties of transfer operators: proof of Theorem 5.4.1 (i)
210(3)
5.6.1 Continuity and analyticity
210(1)
5.6.2 Analyticity in z
211(2)
5.7 Real Hilbert metric estimates
213(5)
5.7.1 General estimates
213(3)
5.7.2 Real cones invariances and diameter of image estimates
216(2)
5.8 Comparison of real and complex operators
218(3)
5.9 Complex image diameter estimates
221(2)
5.10 Real RPF triplets and Gibbs measures
223(3)
5.11 Complex Gibbs functionals
226(2)
5.12 The largest characteristic exponents
228(2)
5.13 Extension to the unbounded case for minimal systems
230(1)
6 Application to random complex integral operators
231(12)
6.1 Integral operators
231(2)
6.2 Real and complex cones
233(1)
6.3 The RPF theorem for integral operators
234(1)
6.4 Properties of cones: proof of Theorem 6.2.1
235(2)
6.5 Real cones: invariance and diameter of image estimates
237(2)
6.6 Comparison of real and complex operators
239(1)
6.7 Complex image diameter estimates
240(3)
7 Limit theorems for processes in random environment
243(26)
7.1 The "conventional" case: preliminaries and main results
243(3)
7.1.1 Self normalized Berry-Esseen theorem
244(1)
7.1.2 Local limit theorem
244(2)
7.2 Pressure near 0
246(5)
7.3 A fiberwise (self normalized) Berry-Esseen theorem: proof
251(2)
7.4 A fiberwise local (central) limit theorem: proof
253(4)
7.4.1 Characteristic functions estimates for small t's
253(1)
7.4.2 Characteristic functions estimates for large t's: covering maps
254(1)
7.4.3 Characteristic functions estimates for large t's: Markov chains
255(2)
7.4.4 An extension
257(1)
7.5 Nonconventional limit theorems for processes in random environment
257(12)
7.5.1 Central limit theorem
257(6)
7.5.2 Local limit theorem
263(6)
Appendix
269(10)
Appendix A Real and complex cones
271(1)
A.1 Real cones and real Hilbert metrics
271(1)
A.2 Complex cones and complex Hilbert metrics
272(7)
A.2.1 Basic notions
272(1)
A.2.2 The canonical complexification of a real cone
273(1)
A.2.3 Apertures and contraction properties
274(1)
A.2.4 Comparison of real and complex operators
275(1)
A.2.5 Further properties of complex dual cones
276(3)
Bibliography 279(4)
Index 283
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