High Dimensional Probability VI
The Banff Volume
1st Edition
Published on 19. April 2013
XIII, 374 pages
978-3-0348-0490-5 (ISBN)
Description
This is a collection of papers by participants at High Dimensional Probability VI Meeting held from October 9-14, 2011 at the Banff International Research Station in Banff, Alberta, Canada.
High Dimensional Probability (HDP) is an area of mathematics that includes the study of probability distributions and limit theorems in infinite-dimensional spaces such as Hilbert spaces and Banach spaces. The most remarkable feature of this area is that it has resulted in the creation of powerful new tools and perspectives, whose range of application has led to interactions with other areas of mathematics, statistics, and computer science. These include random matrix theory, nonparametric statistics, empirical process theory, statistical learning theory, concentration of measure phenomena, strong and weak approximations, distribution function estimation in high dimensions, combinatorial optimization, and random graph theory.
The papers in this volume show that HDP theory continues to develop new tools, methods, techniques and perspectives to analyze the random phenomena. Both researchers and advanced students will find this book of great use for learning about new avenues of research.
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Content
- Intro
- Contents
- Preface
- Inequalities and Convexity:
- Limit Theorems:
- Stochastic Processes:
- Random Matrices and Applications:
- High Dimensional Statistics:
- Participants
- Dedication
- Part I Inequalities and Convexity
- Bracketing Entropy of High Dimensional Distributions
- 1. Introduction
- 2. Bracketing entropy estimate
- 3. Bounding bracketing entropy using metric entropy
- Acknowledgement
- References
- Slepian's Inequality, Modularity and Integral Orderings
- 1. Introduction
- 2. Slepian's inequality
- 3. Integral orderings
- 4. Modular orderings
- References
- A More General Maximal Bernstein-type Inequality
- 1. Introduction
- Acknowledgment
- References
- Maximal Inequalities for Centered Norms of Sums of Independent Random Vectors
- 1. Introduction and main results
- 2. Proofs
- 3. Example
- References
- A Probabilistic Inequality Related to Negative Definite Functions
- 1. Introduction
- 2. Main result
- 3. A relation to random processes
- 4. Relation to bifractional Brownian motion
- 5. A counterexample
- Acknowledgement
- References
- Optimal Re-centering Bounds, with Applications to Rosenthal-type Concentration of Measure Inequalities
- 1. Introduction
- 2. Summary and discussion
- 3. Application: Rosenthal-type concentration inequalities for separately Lipschitz functions on product spaces
- 4. Proofs
- Acknowledgment
- References
- Strong Log-concavity is Preserved by Convolution
- 1. Log-concavity and ultra-log-concavity for discrete distributions
- 2. Log-concavity and strong-log-concavity for continuous distributions on R
- 3. Appendix: strong convexity and strong log-concavity
- Acknowledgment
- References
- On Some Gaussian Concentration Inequality for Non-Lipschitz Functions
- 1. Introduction
- 2. The result
- 3. Application to U-statistics
- References
- Part II Limit Theorems
- Rates of Convergence in the Strong Invariance Principle for Non-adapted Sequences. Application to Ergodic Automorphisms of the Torus
- 1. Introduction and notations
- 2. ASIP with rates for ergodic automorphisms of the torus
- 3. Probabilistic results
- 4. Proof of Theorem 2.1
- 4.1. Preparatory material
- 4.2. End of the proof of Theorem 2.1
- 5. Appendix
- References
- On the Rate of Convergence to the Semi-circular Law
- 1. Introduction
- 2. Bounds for the Kolmogorov distance between distribution functions via Stieltjes transforms
- 3. Large deviations I
- 4. Bounds for │mn(z)│
- 5. Large deviations II
- 6. Stieltjes transforms
- 7. Proof of Theorem 1.1
- 8. Proof of Theorem 1.2
- Acknowledgement
- References
- Empirical Quantile CLTs for Time-dependent Data
- 1. Introduction
- 2. Statement of results
- 3. The proof of Theorem 1: Vervaat's approach
- 3.1. Notation and some lemmas
- 3.2. Applying Lemma 2 to obtain an empirical quantile CLT
- 3.3. Proof of Theorem 1
- 4. Proof of Theorem 2
- 4.1. Gaussian process empirical CLT's over C
- 4.2. Compound Poisson process empirical CLT's over C
- 4.3. Empirical process CLT's over C for other independent increment processes and martingales
- References
- Asymptotic Properties for Linear Processes of Functionals of Reversible or Normal Markov Chains
- 1. Introduction
- 2. Definitions, background and results
- 3. Applications
- 3.1. Application to a Metropolis Hastings Markov chain
- 3.2. Linear process of instantaneous functions of a Gaussian sequence
- 3.3. Application to random walks on compact groups
- 4. Proofs
- 4.1. Preliminary general results
- 4.2. Normal and reversible Markov chains
- 5. Appendix
- Acknowledgement
- References
- Part III Stochastic Processes
- First Exit of Brownian Motion from a One-sided Moving Boundary
- 1. Introduction
- 2. Proof
- 3. Further remarks
- References
- On Lévy's Equivalence Theorem in Skorohod Space
- 1. Introduction
- 1.1. Definitions and notations
- 2. Lévy's Equivalence Theorem for D([0, 1]
- E)
- References
- Continuity Conditions for a Class of Second-order Permanental Chaoses
- 1. Introduction
- 2. Continuity
- 3. Proof of Theorem 1.1
- 4. Domination by the second moment
- References
- Part IV Random Matrices and Applications
- On the Operator Norm of Random Rectangular Toeplitz Matrices
- 1. Introduction
- 2. Proof of Theorem 1.1
- 3. Proof of Proposition 1.2
- 4. Appendix
- References
- Edge Fluctuations of Eigenvalues of Wigner Matrices
- 1. Introduction
- 2. Global moderate deviations at the edge of the spectrum
- 3. Local moderate deviations at the edge of the spectrum
- 4. Universal local moderate deviations near the edge
- 5. Universal global moderate deviations near the edge
- 6. Further random matrix ensembles
- Acknowlegement
- References
- On the Limiting Shape of Young Diagrams Associated with Inhomogeneous Random Words
- 1. Introduction
- 2. Generalized traceless GUE
- 3. Young diagrams and inhomogeneous random words
- 4. The Poissonized Word Problem
- 5. Appendix
- Acknowledgment
- References
- Part V High Dimensional Statistics
- Low Rank Estimation of Similarities on Graphs
- 1. Introduction
- 2. Main results
- 3. Proofs
- References
- Sparse Principal Component Analysis with Missing Observations
- 1. Introduction
- 2. Tools and definitions
- 3. Main results for sparse PCA with missing observations
- 4. Proofs
- 4.1. Proof of Propositions 2 and 3
- 4.2. Proof of Theorems 1 and 2
- 4.3. Proof of Lemma 2
- 4.4. Proof of Theorem 4
- 4.5. Proof of Lemma 4
- 4.6. Proof of Lemma 5
- 4.7. Proof of Fact 1
- 4.8. Proof of Fact 2
- 4.9. Bounding of the moment E[(?T Z?)2]
- References
- High Dimensional CLT and its Applications
- 1. Introduction
- 2. Main results
- 3. Applications
- Quantile approximation justification
- Application
- GOF test
- Numerical results
- Comments for Table 3.1
- 4. Proofs
- References
