Marginal and Functional Quantization of Stochastic Processes

Preface
Contents
Notation Index
	Other Notations
	About Sequences
Part I Basics and Marginal Quantization
	Chapter 1 Optimal and Stationary Quantizers
		1.1 What is Functional Quantization?
		1.2 Optimal Quantizers and Quantization Error
		1.3 Stationary Quantizers
			Stationarity for random vectors in a Hilbert space
			Stationarity for stochastic processes
		1.4 Pathwise Regularity of Stationary Quantizers in ???????? -Spaces and Examples
		1.5 Increments of Quantization Errors and Micro-Macro Inequalities
		1.6 Greedy Quantization
		1.7 Application: Quantization-Based Characterizations of W????-Convergence
			Hilbert setting
			Finite-dimensional setting:
		Comments
	Chapter 2 The Finite-Dimensional Setting I
		2.1 Sharp Asymptotics of the Quantization Errors and the Point Density Measure
			Sharp asymptotics
			The point density measure
			Universal nonasymptotic bounds and sharp asymptotics
		2.2 Exact Asymptotics of the Increments of Quantization Errors
		2.3 Distortion Mismatch and Distribution Mismatch
			The lower estimate
			The upper estimate
			Pure distortion mismatch
			Pure distribution mismatch
		2.4 Geometry of Optimal Quantizers
			Radial exponential tails
			Radial polynomial tails
			Quantization radius and random quantization
		2.5 Geometry of Rate-Optimal Quantizers
		2.6 Local Behaviour of Optimal Quantizers
			Distance of codewords, inradius and circumradius of Voronoi cells
			Distributions with connected compact support
			The local quantization behaviour in compact subsets of the interior of the support
			Sharp results in dimension 1
			Distributions with radial tails
		Comments
	Chapter 3 The Finite-Dimensional Setting II
		3.1 Random Quantization
			Mean ???????? -quantization error
			Almost sure behaviour of ???????? -quantization errors
		3.2 Empirical Quantization
			Nonasymptotic bounds
		3.3 Greedy Quantization
			Universal nonasymptotic bounds and rate-optimality
			Distortion mismatch and distribution mismatch
			Pure distortion mismatch and universal nonasymptotic bounds
			Geometric features
			Further remarks (greedy quantization versus quasi-Monte Carlo method)
		Comments
Part II Functional Quantization
	Chapter 4 Functional Quantization, Small Ball Probabilities, Metric Entropy and Series Expansions for Gaussian Processes
		4.1 Functional Quantization and Small Ball Probabilities
			Upper bound and random quantization
			Lower bound
			Synthesis
		4.2 Functional Quantization and Metric Entropy
		4.3 Applications and Examples: Exact Rates of Decay of Quantization Errors
			Mean regular processes
			Stationary processes
			Sheets
			Smooth processes
			R????-valued processes
		4.4 Functional Quantization, Expansions and Parseval Frames
			Admissibility and frames
			Continuous Gaussian processes
			A criterion for rate-optimal expansions
			l-norms
			????2(P)-continuous Gaussian processes
			Stationary Gaussian processes and Gaussian processes with stationary increments
			Functional quantization and expansions
		4.5 Product Quantization, Constructive Asymptotic Optimality and Distortion Mismatch
			Scalar product quantization and distortion mismatch
			Constructive asymptotic optimality
		Comments
	Chapter 5 Spectral Methods for Gaussian Processes
		5.1 Optimal and Stationary Quantizers
			The quadratic case
			Finite-dimensional subproblems
			The Gaussian case
			Comparison of rates
		5.2 Shannon Entropy
			The Kolmogorov–Pinsker–Ihara formula and a source coding theorem
			Two quantization constants
			Comparison of rates
		5.3 Critical Dimension and Sharp Asymptotics of the Quantization Errors
			Regularly varying eigenvalues, quadratic case
			Upper bound
			Lower bound
			Synthesis
			Entropy and small ball probabilities
			Back to the conjecture(s)
			The case
		5.4 Shannon Entropy and Strong Equivalence of Moments
			Proof of Theorem 5.4.1
			Exponential decay of eigenvalues
		5.5 Applications and Examples
		5.6 Constructive Asymptotic Optimality, Product Quantization and Distortion Mismatch
			Known eigensystem
			Unknown eigenvectors
		Comments
	Chapter 6 Geometry of Optimal and Rate-Optimal Quantizers for Gaussian Processes
		6.1 Quantization Ball and Radius of Optimal Quantizers
		6.2 Increments of the Quantization Errors
			Covering radius
			Quantization ball
		6.3 Proofs of the Results of Section 6.1
		6.4 Quantization Ball and Radius of Rate-Optimal Quantizers
		Comments
	Chapter 7 Mean Regular Processes
		7.1 Product Quantization, Universal Upper Bounds for Quantization Error Rates and Constructive Asymptotic Optimality
			Product quantization and upper bound for quantization error rates
			Constructive asymptotic optimality
			A nonconstructive link
		7.2 Brownian Diffusion Processes
			Itô processes
			Brownian diffusions
			Constructive rate-optimal quantization
			Product quantization based on the Euler scheme
			Constructive asymptotic optimality based on the Euler scheme
		7.3 Lévy Processes
			Moment estimates, quantization rates for general Lévy processes without Brownian component
			An exact rate for Lévy processes with a Brownian component
			Subordinated Lévy processes
			Regularly varying Lévy tails, ???? ≥ ????∗ (????)
			Constructive asymptotic optimality
			Final remark on R????-valued Lévy processes
		Comments
Part III Algorithmic Aspects and Applications
	Chapter 8 Optimal Quantization From the Numerical Side (Static)
		8.1 From the Master Equation to Grid Optimization
			Zero search procedures
		8.2 Fixed Point Procedures (Lloyd’s Family)
			Nearest neighbour search
		8.3 The One-Dimensional Quadratic Case
			Back to the fixed point Lloyd algorithm (and its acceleration)
			Gradient descent: the “mean-field/batch” CLVQ algorithm
			Newton–Raphson algorithm (fast quantization for absolutely continuous distributions)
			Practitioner’s corner (splitting method)
		8.4 Uniqueness of ????????-Stationary Quantizers for log-Concave Distributions
		8.5 Greedy Algorithms
			l-dimensional setting
			Deterministic algorithm in the two-dimensional case (greedy Lloyd algorithm)
		Comments
	Chapter 9 Applications: Quantization-Based Cubature Formulas
		9.1 Cubature Formulas for Expectations and Conditional Expectations
			Cubature formulas based on stationary quantizations
			Quantization-based cubature formulas for conditional expectations
			Quantization versus worst case Monte Carlo method
		9.2 Cubature: Rates
			Finite-dimensional setting: ???? = R????
			Functional setting: ???? = ????(T)
			Hilbert setting
		9.3 Richardson–Romberg Extrapolation
			The 1-dimensional case
			General case (heuristics)
		9.4 Quantization-Based Variance Reduction Method
			A hybrid quantization-Monte Carlo method
			Universal stratified sampling
			A(n optimal) quantization-based universal stratification: a minimax approach
		Comments
	Chapter 10 Quantization-Based Numerical Schemes
		10.1 Standard Marginal Quantization of an R????-Valued Markov Chain
			Monte Carlo calibration of a quantization tree
			The case of smooth transitions or how to parallelize the transition weights computation
			The one-dimensional setting
		10.2 Example of Application: Quantization-Based Algorithm for Discrete Time Optimal Stopping
			Euler scheme of a Brownian diffusion with Lipschitz coefficients
			Brownian quantization tree
		10.3 Practitioner’s Corner: How to Calibrate a Quantization Tree (Example)
		10.4 Recursive Markovian Quantization of a Markov Chain
			Strong approximation, Setting I
			Strong approximation, Setting II
			Embedded optimization of the Markovian quantization tree: grid optimization and weight computation
			Back to optimal stopping: approach by recursive quantization
			Other applications
		10.5 Weak Error Rate for Recursive Quantization (Smooth Functions, Setting I)
			Application to numerical schemes of diffusions
		10.6 Standard Versus Recursive Quantization-Based Schemes?
		10.7 Further Numerical Applications of Optimal Quantization(s)
		Comments
Appendices
	Appendix A Radon Random Vectors, Stochastic Processes and Inequalities
		A.1 Radon Random Vectors
		A.2 Radon Gaussian Random Vectors
		A.3 Stochastic Processes as Random Vectors
			???????? -spaces
			C-spaces
		A.4 Conditional Expectation
	Appendix B Miscellany
		B.1 Hausdorff Metric
		B.2 Regular Variation
		B.3 Sequences of Real Numbers
		B.4 Hardy–Littlewood Maximal Functions and Volume Control
References
Index