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Functional Gaussian Approximation for Dependent Structures [Kõva köide]

(University of Leicester, Professor), (Professor, University of Cincinnati), (Professor, Université Paris-Est Marne-La-Vallée)
  • Formaat: Hardback, 496 pages, kõrgus x laius x paksus: 237x165x29 mm, kaal: 940 g
  • Sari: Oxford Studies in Probability
  • Ilmumisaeg: 05-Mar-2019
  • Kirjastus: Oxford University Press
  • ISBN-10: 019882694X
  • ISBN-13: 9780198826941
Teised raamatud teemal:
Functional Gaussian Approximation for Dependent StructuresSuurem pilt
  • Formaat: Hardback, 496 pages, kõrgus x laius x paksus: 237x165x29 mm, kaal: 940 g
  • Sari: Oxford Studies in Probability
  • Ilmumisaeg: 05-Mar-2019
  • Kirjastus: Oxford University Press
  • ISBN-10: 019882694X
  • ISBN-13: 9780198826941
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780198826941.html
Märksõnad:
Functional Gaussian Approximation for Dependent Structures develops and analyses mathematical models for phenomena that evolve in time and influence each another. It provides a better understanding of the structure and asymptotic behaviour of stochastic processes.

Two approaches are taken. Firstly, the authors present tools for dealing with the dependent structures used to obtain normal approximations. Secondly, they apply normal approximations to various examples. The main tools consist of inequalities for dependent sequences of random variables, leading to limit theorems, including the functional central limit theorem and functional moderate deviation principle. The results point out large classes of dependent random variables which satisfy invariance principles, making possible the statistical study of data coming from stochastic processes both with short and long memory.

The dependence structures considered throughout the book include the traditional mixing structures, martingale-like structures, and weakly negatively dependent structures, which link the notion of mixing to the notions of association and negative dependence. Several applications are carefully selected to exhibit the importance of the theoretical results. They include random walks in random scenery and determinantal processes. In addition, due to their importance in analysing new data in economics, linear processes with dependent innovations will also be considered and analysed.

Arvustused

This book is welcome. It gathers the authors' contribution to an important topic as well as many recent significant results obtained by other master researchers in a central domain of probability theory. * Dominique Lépingle, zbMATH *

List of Notations
xiii
1 Introduction to Stochastic Processes
1(26)
1.1 Introduction
1(7)
1.1.1 Discussion
1(3)
1.1.2 Examples
4(4)
1.2 Stationary Sequences
8(10)
1.2.1 Definitions
8(2)
1.2.2 Constructions
10(2)
1.2.3 Ergodicity
12(2)
1.2.4 Projective Decomposition
14(1)
1.2.5 Spectral Density for Second Order Real-valued Stationary Stochastic Processes
15(3)
1.3 Introduction to Convergence of Stochastic Processes
18(9)
1.3.1 Useful Facts Concerning Convergence in Distribution
18(1)
1.3.2 Weak Convergence of the Partial Sums Process
19(2)
1.3.3 Maximal Moment Inequalities
21(2)
1.3.4 Moderate Deviations
23(4)
PART I PROBABILISTIC TOOLS
2 Moment Inequalities and Gaussian Approximation for Martingales
27(35)
2.1 Definitions and Properties
27(1)
2.2 Maximal and Moment Inequalities for Martingales
28(11)
2.2.1 Doob's Maximal Inequality
28(4)
2.2.2 Moment Inequalities for Martingales
32(3)
2.2.3 Exponential Inequalities for Martingales
35(4)
2.3 Central Limit Theorem for Triangular Arrays of Martingales
39(7)
2.4 Functional Central Limit Theorem for Triangular Arrays of Martingales
46(6)
2.5 Moderate Deviations for Martingales
52(10)
3 Moment Inequalities via Martingale Methods
62(33)
3.1 Analysis of the Variance of Partial Sums in the Stationary Setting: The Dyadic Induction
62(5)
3.2 Burkholder-type Inequalities
67(5)
3.2.1 Burkholder-type Inequalities via Maxwell-Woodroofe type Characteristics
67(2)
3.2.2 Burkholder-type Inequalities for Non-Stationary Sequences via Projective Criteria
69(1)
3.2.3 A Maximal Inequality a la Doob for Adapted Sequences
70(2)
3.3 A Rosenthal-type Inequality for Stationary Sequences
72(1)
3.4 Maximal Exponential Inequalities
73(1)
3.5 Proofs
74(17)
3.5.1 Proofs of the Results of Section 3.2
74(11)
3.5.2 Proof of Theorem 3.17
85(5)
3.5.3 Proofs of the Results of Section 3.4
90(1)
3.6 Facts about Subadditive Sequences
91(4)
4 Gaussian Approximation via Martingale Methods
95(50)
4.1 Martingale Approximations
95(9)
4.1.1 General Martingale Approximations
95(3)
4.1.2 Stationary Martingale Approximation in L2
98(6)
4.2 The Central Limit Theorem for Stationary Sequences in L1 or in L2
104(6)
4.3 On the Functional Central Limit Theorem for Stationary Sequences in L2
110(3)
4.4 On the Functional Central Limit Theorem for Non-Stationary Sequences in L2
113(18)
4.4.1 Statements
113(3)
4.4.2 Proofs
116(15)
4.5 Toward a More General Normalization
131(8)
4.5.1 Proof of Theorem 4.18
136(3)
4.6 Moderate Deviations
139(6)
5 Dependence Coefficients for Sequences
145(20)
5.1 Traditional Mixing Coefficients
145(14)
5.1.1 Definitions
145(2)
5.1.2 Examples of Mixing Processes
147(4)
5.1.3 Coupling Properties of the Mixing Coefficients
151(7)
5.1.4 Examples of Non-Mixing Processes
158(1)
5.2 Weak Dependence Coefficients
159(6)
5.2.1 Definitions
160(2)
5.2.2 The Weak Dependence Coefficients in the Markov Chains Setting
162(1)
5.2.3 Examples
162(3)
6 Moment Inequalities and Gaussian Approximation for Mixing Sequences
165(68)
6.1 Probability and Moment Inequalities for p-mixing Sequences
165(24)
6.1.1 Analysis of the Variance: A Twist of the Dyadic Induction
166(9)
6.1.2 An Extension of the Variance Inequality
175(4)
6.1.3 The Maximal Version of the Variance Inequality under p-mixing
179(3)
6.1.4 Rosenthal-Type Inequality under p-mixing
182(7)
6.2 Probability and Moment Maximal Inequalities under mixing
189(7)
6.3 Moment Inequalities for a-dependent Sequences
196(10)
6.3.1 Burkholder-type Inequalities
197(4)
6.3.2 Rosenthal-type Inequality
201(4)
6.3.3 Comments
205(1)
6.4 Invariance Principle for Strong Mixing Sequences
206(8)
6.5 Gaussian Approximation under Lindeberg's Condition
214(15)
6.5.1 A General Result
215(9)
6.5.2 Application: CLT for L1-mixing Triangular Arrays under Lindeberg's Condition
224(5)
6.6 Invariance Principle for Stationary p-mixing Sequences
229(2)
6.7 Central Limit Theorem for L7-mixing Sequences and Discussion on Ibragimov Conjecture
231(2)
7 Weakly Associated Random Variables: L2 -Bounds and Approximation by Independent Structures
233(18)
7.1 Definition of the Weak Associated Coefficients between Two Random Variables
233(3)
7.2 Approximation by Independent Pair
236(4)
7.2.1 Weak Negative Dependence
236(2)
7.2.2 Weak Positive Dependence
238(1)
7.2.3 The Weak Martingale Coefficient
238(2)
7.3 Preservation of the Weak Negative or Positive Dependence under Convolution
240(1)
7.4 L2-bounds via Interlaced Quantities
241(5)
7.4.1 Definitions of the Linear Negative or Positive Dependence Coefficients
242(1)
7.4.2 Estimates of the Variance under Near Association Type Conditions
242(4)
7.5 Approximation Inequalities for n Variables
246(5)
7.5.1 A First Approximation Inequality
246(2)
7.5.2 A Second Approximation Inequality
248(3)
8 Maximal Moment Inequalities for Weakly Negatively Dependent Variables
251(26)
8.1 General Moment Inequalities for Weakly Negatively Dependent Variables
251(3)
8.2 A Khintchine-Marcinkiewicz-Zygmund type Inequality
254(2)
8.3 Rosenthal Moment Inequalities
256(4)
8.3.1 Definition of r-negatively Dependent Vectors
257(1)
8.3.2 Rosenthal-type Moment Inequality for the Partial Sums
258(2)
8.4 Maximal Moment Inequalities for Partial Sums
260(4)
8.5 The Weak Law of the Large Numbers Type Arguments
264(5)
8.5.1 Convergence for Near-Stationary Sequences
265(1)
8.5.2 Convergence under rND Condition
266(1)
8.5.3 Rnd-generalizauon Principle
267(1)
8.5.4 Beurling-Malliavin Density of Weakly Negatively Dependent Random Sequences
268(1)
8.6 A Useful Fact Concerning the rND Condition
269(8)
9 Gaussian Approximation under Asymptotic Negative Dependence
277(28)
9.1 General Construction and Tightness
277(8)
9.1.1 Univariate Construction and Assumptions
277(1)
9.1.2 Condition (A)
278(1)
9.1.3 Convention
278(1)
9.1.4 Multivariate Construction and Multivariate Condition (A)
278(1)
9.1.5 Tightness
279(3)
9.1.6 Some Properties of rND Processes
282(3)
9.2 Convergence to a Gaussian Process with Independent Increments
285(6)
9.2.1 Univariate Invariance Principle
285(3)
9.2.2 Multivariate Invariance Principle
288(3)
9.3 Convergence to a Diffusion Process with Deterministic Time Varying Volatility
291(5)
9.4 CLT and IP for Stationary Sequences of Asymptotic Negatively or Positively Dependent Variables
296(9)
PART II EXAMPLES
10 Examples of Stationary Sequences with Approximate Negative Dependence
305(14)
10.1 Determinantal Point Processes and Their Perturbations
305(4)
10.2 Displaced Point Processes
309(2)
10.3 Exchangeable Processes via the rND Property
311(8)
10.3.1 Weighted Empirical Processes
314(1)
10.3.2 An Example with Convergence to a Non-Gaussian Process for Exchangeable Negatively Dependent Variables
315(2)
10.3.3 Exchangeable Determinantal Point Processes
317(2)
11 Stationary Sequences in a Random Time Scenery
319(26)
11.1 Sampling by a Shifted Markov Chain
319(19)
11.1.1 Introduction
319(1)
11.1.2 The Chung Chain, the Renewal Chain and the Model
319(2)
11.1.3 Notations and Properties of the "Renewal" Chain and of the Sampled Random Scenery
321(3)
11.1.4 Martingale Difference Scenery
324(3)
11.1.5 Stationary Time Scenery
327(2)
11.1.6 Invariance Principle for a Process Sampling by the Shifted Renewal Markov Chain
329(9)
11.2 Projective Approach for RWRS
338(7)
12 Linear Processes
345(37)
12.1 Linear Processes with Short Memory
345(5)
12.2 Functional CLT using Coboundary Decomposition
350(1)
12.3 Toward Linear Processes with Long Memory
351(10)
12.3.1 CLT for Linear Statistics with Dependent Innovations via Martingale Approximation
352(9)
12.4 Invariance Principle for Linear Processes
361(6)
12.4.1 Construction of the Counterexample
362(2)
12.4.2 Finite-dimensional Distributions
364(2)
12.4.3 Tightness
366(1)
12.5 IP for Linear Statistics with Weakly Associated Innovations
367(5)
12.5.1 The Case of Asymptotically Negative Dependent Innovations
367(1)
12.5.2 The Case of Long-Range Dependent Statistics of Stationary Perturbed Determinantal Point Processes
368(4)
12.6 Discrete Fourier Transform and Periodogram
372(10)
12.6.1 A CLT for Almost All Frequencies
373(8)
12.6.2 Examples
381(1)
13 Random Walk in Random Scenery
382(23)
13.1 Introduction and Main Results
382(8)
13.1.1 On the Central Limit Behavior of Zn
383(1)
13.1.2 On the Functional form of the Central Limit Theorem
384(6)
13.2 Properties of the Random Walk
390(12)
13.3 CLT for Random Walk Self-intersections--Non-linear Normalizer
402(3)
14 Reversible Markov Chains
405(23)
14.1 Introduction
405(1)
14.2 Functional Central Limit Theorem for Reversible Markov Chains under Normalization √n
406(5)
14.3 Maximal Inequalities
411(5)
14.4 Invariance Principle for Reversible Markov Chains under General Normalization
416(7)
14.4.1 Introduction and Result
416(1)
14.4.2 Proof of Theorem 14.9
417(6)
14.4.3 Proof of Corollary 14.10
423(1)
14.5 Application to a Metropolis-Hastings Algorithm
423(5)
15 Functional Central Limit Theorem for Empirical Processes
428(10)
15.1 Tightness Criteria
429(1)
15.2 Applications to Dependent Sequences
430(8)
15.2.1 P-mixing Sequences
431(1)
15.2.2 A-dependent Sequences
432(1)
15.2.3 P-dependent Sequences
433(5)
16 Application to the Uniform Laws of Large Numbers for Dependent Processes
438(10)
16.1 The Case of Absolutely Regular Sequences
439(1)
16.2 The Case of φ-mixing Sequences
439(5)
16.3 The Case of Strong Mixing Sequences
444(4)
17 Examples and Counterexamples
448(17)
17.1 The Renewal-Type Markov Chain Example
448(8)
17.1.1 L2 Analysis for the Renewal Chain Example
449(5)
17.1.2 Lp Analysis for the Renewal Chain Example
454(2)
17.2 Telescopic Examples
456(9)
17.2.1 CLT Does NOT Imply IP for Stationary Sequences under any Normalization
457(4)
17.2.2 Linear Asymptotic Variance and CLT Do Not Imply IP
461(4)
References 465(12)
Subject Index 477
Florence Merlevède is Professor at the Laboratory of Analysis and Applied Mathematics of the University of Paris-Est. Her main research interests are in moment inequalities, deviations probability inequalities, limit theorems for partial sums associated to dependent processes, empirical processes but include also dynamical systems and random matrices

Magda Peligrad is a distinguished Taft Professor in the Department of Mathematical Sciences of the University of Cincinnati whose area of expertise is Probability Theory and Stochastic Processes. Her research deals with dependent structures and covers various aspects of modelling the dependence, maximal inequalities, and limit theorems. Her research was rewarded by numerous National Science Foundation, National Security Agency, and Taft research center grants. In 1995 she was elected as fellow of the Institute of Mathematical Statistics, in 2003 she received the title of Taft Professor at the University of Cincinnati and in 2010 her contributions to Probability theory were recognized in a meeting held in her honor in Paris, France

Sergey Utev is Professor in the Department of Mathematics of the University of Leicester. His area of expertise covers many aspects of Probability theory and their applications. In particular he wrote many important papers concerning mathematical inequalities and their applications, Quantum probability and stochastic comparisons, Stochastic processes with applications to Financial Mathematics, Actuarial Sciences and Epidemics