This volume contains two of the three lectures that were given at the 33rd Probability Summer School in Saint-Flour (July 6-23, 2003). Amir Dembo’s course is devoted to recent studies of the fractal nature of random sets, focusing on some fine properties of the sample path of random walk and Brownian motion. In particular, the cover time for Markov chains, the dimension of discrete limsup ran…
The Haar System: Basic Facts and Classical Results -- Projections, Isomorphisms and Interpolation -- Combinatorics of Colored Dyadic Intervals -- Martingale H1 Spaces -- Isomorphic Invariants for H1 -- Atomic H1 Spaces.This book presents a thorough and self-contained presentation of H¹ and its known isomorphic invariants, such as the uniform approximation property, the dimension conjecture, an…
Introduction -- 1. Preliminaries on Markov-Feller Operators -- 2. The KBBY Decomposition -- 3. Unique Ergodicity -- 4. Equicontinuity -- Bibliography -- Index.In this book invariant probabilities for a large class of discrete-time homogeneous Markov processes known as Feller processes are discussed. These Feller processes appear in the study of iterated function systems with probabilities, conv…
The Theory of Stochastic Processes -- Fundamentals of Probability -- Stochastic Processes -- The Itô Integral -- Stochastic Differential Equations -- The Applications of Stochastic Processes -- Applications to Finance and Insurance -- Applications to Biology and Medicine.This concisely written book is a rigorous and self-contained introduction to the theory of continuous-time stochastic proces…
Stochastic Methods in Statistical Physics -- Coarse-Graining Techniques for (Random) Kac Models -- Euclidean Gibbs Measures of Quantum Crystals: Existence, Uniqueness and a Priori Estimates -- Some Jump Processes in Quantum Field Theory -- Gibbs Measures on Brownian Paths: Theory and Applications -- Spectral Theory for Nonstationary Random Potentials -- A Survey of Rigorous Results on Random Sc…
The Construction, and Other General Results -- Some Basic Tools -- Spin Systems -- Stochastic Ising Models -- The Voter Model -- The Contact Process -- Nearest-Particle Systems -- The Exclusion Process -- Linear Systems with Values in [0, ?)s.From the reviews "[...] This book presents a complete treatment of a new class of random processes, which have been studied intensively during the last fi…
Main Definitions and Notations -- Main Definitions and Notations -- State Inference -- Filtering and Smoothing Recursions -- Advanced Topics in Smoothing -- Applications of Smoothing -- Monte Carlo Methods -- Sequential Monte Carlo Methods -- Advanced Topics in Sequential Monte Carlo -- Analysis of Sequential Monte Carlo Methods -- Parameter Inference -- Maximum Likelihood Inference, Part I: Op…
CUSUM Procedure -- Change-Point Estimation -- Confidence Interval for Change-Point -- Inference for Post-Change Mean -- Estimation After False Signal -- Inference with Change in Variance -- Sequential Classification and Segmentation -- An Adaptive CUSUM Procedure -- Dependent Observation Case -- Other Methods and Remarks.This monograph is the first to systematically study the bias of estimators…
Gradient Flow in Metric Spaces -- Curves and Gradients in Metric Spaces -- Existence of Curves of Maximal Slope and their Variational Approximation -- Proofs of the Convergence Theorems -- Uniqueness, Generation of Contraction Semigroups, Error Estimates -- Notation -- Gradient Flow in the Space of Probability Measures -- Preliminary Results on Measure Theory -- The Optimal Transportation Probl…
Overview of Geometric Data Analysis (‘Overview’) -- Correspondence Analysis -- Euclidean Cloud -- Principal Component Analysis -- Multiple Correspondence Analysis (MCA) -- Structured Data Analysis -- Stability of a Euclidean Cloud -- Inductive Data Analysis -- Research Case Studies -- Mathematical Bases.Geometric Data Analysis (GDA) is the name suggested by P. Suppes (Stanford University) t…