Shalizi C.R., Kontorovich A. Almost none of the theory of stochastic processes

Table of Contents Comprehensive List of Definitions, Lemmas, Propositions, Theorems, Corollaries, Examples and Exercises Preface I Stochastic Processes in General Basics So, What Is a Stochastic Process? Random Functions Exercises Building Processes Finite-Dimensional Distributions Consistency and Extension Building Processes by Conditioning Probability Kernels Extension via Recursive Conditioning Exercises II One-Parameter Processes in General One-Parameter Processes One-Parameter Processes Operator Representations of One-Parameter Processes Exercises Stationary Processes Kinds of Stationarity Strictly Stationary Processes and Measure-Preserving Transformations Exercises Random Times Reminders about Filtrations and Stopping Times Waiting Times Kac's Recurrence Theorem Exercises Continuity Kinds of Continuity for Processes Why Continuity Is an Issue Separable Random Functions Separable Versions Measurable Versions Cadlag Versions Continuous Modifications Exercises III Markov Processes Markov Processes The Correct Line on the Markov Property Transition Probability Kernels The Markov Property Under Multiple Filtrations Exercises Markov Characterizations Markov Sequences as Transduced Noise Time-Evolution (Markov) Operators Exercises Markov Examples Probability Densities in the Logistic Map Transition Kernels and Evolution Operators for the Wiener Process Lévy Processes and Limit Laws Exercises Generators Exercises Strong Markov, Martingales The Strong Markov Property Martingale Problems Exercises Feller Processes Markov Families Feller Processes Exercises Convergence of Feller Processes Weak Convergence of Processes with Cadlag Paths (The Skorokhod Topology) Convergence of Feller Processes Approximation of Ordinary Differential Equations by Markov Processes Exercises Convergence of Random Walks The Wiener Process is Feller Convergence of Random Walks Approach Through Feller Processes Direct Approach Consequences of the Functional Central Limit Theorem Exercises IV Diffusions and Stochastic Calculus Diffusions and the Wiener Process Diffusions and Stochastic Calculus Once More with the Wiener Process and Its Properties Wiener Measure; Most Continuous Curves Are Not Differentiable Stochastic Integrals A Heuristic Introduction to Stochastic Integrals Integrals with Respect to the Wiener Process Some Easy Stochastic Integrals, with a Moral dW WdW Itô's Formula Stratonovich Integrals Martingale Characterization of the Wiener Process Martingale Representation Exercises Stochastic Differential Equations Brownian Motion, the Langevin Equation, and Ornstein-Uhlenbeck Processes Solutions of SDEs are Diffusions Forward and Backward Equations Exercises Spectral Analysis and White Noise Integrals Spectral Representation of Weakly Stationary Procesess White Noise How the White Noise Lost Its Color Small-Noise SDEs Convergence in Probability of SDEs to ODEs Rate of Convergence; Probability of Large Deviations V Ergodic Theory The Mean-Square Ergodic Theorem Mean-Square Ergodicity Based on the Autocovariance Mean-Square Ergodicity Based on the Spectrum Exercises Ergodic Properties and Ergodic Limits General Remarks Dynamical Systems and Their Invariants Time Averages and Ergodic Properties Asymptotic Mean Stationarity Exercises The Almost-Sure Ergodic Theorem Ergodicity and Metric Transitivity Metric Transitivity Examples of Ergodicity Consequences of Ergodicity Deterministic Limits for Time Averages Ergodicity and the approach to independence Exercises Ergodic Decomposition Preliminaries to Ergodic Decompositions Construction of the Ergodic Decomposition Statistical Aspects Ergodic Components as Minimal Sufficient Statistics Testing Ergodic Hypotheses Exercises Mixing Definition and Measurement of Mixing Examples of Mixing Processes Convergence of Distributions Under Mixing A Central Limit Theorem for Mixing Sequences Asymptotic Distributions [[w]] VI Information Theory Entropy and Divergence Shannon Entropy Relative Entropy or Kullback-Leibler Divergence Statistical Aspects of Relative Entropy Mutual Information Mutual Information Function Rates and Equipartition Information-Theoretic Rates Asymptotic Equipartition Typical Sequences Asymptotic Likelihood Asymptotic Equipartition for Divergence Likelihood Results Exercises Information Theory and Statistics [[w]] VII Large Deviations Large Deviations: Basics Large Deviation Principles: Main Definitions and Generalities Breeding Large Deviations IID Large Deviations Cumulant Generating Functions and Relative Entropy Large Deviations of the Empirical Mean in Rd Large Deviations of the Empirical Measure in Polish Spaces Large Deviations of the Empirical Process in Polish Spaces Large Deviations for Markov Sequences Large Deviations for Pair Measure of Markov Sequences Higher LDPs for Markov Sequences The Gärtner-Ellis Theorem The Gärtner-Ellis Theorem Exercises Large Deviations for Stationary Sequences [[w]] Large Deviations in Inference [[w]] Freidlin-Wentzell Theory Large Deviations of the Wiener Process Large Deviations for SDEs with State-Independent Noise Large Deviations for State-Dependent Noise Exercises VIII Measure Concentration [Kontorovich/w] IX Partially-Observable Processes [[w]] Hidden Markov Models [[w]] Stochastic Automata [[w]] Predictive Representations [[w]] X Applications [[w]] XI Appendices [[w]] Real and Complex Analysis [[w]] Real Analysis Complex Analysis General Vector Spaces and Operators [[w]] Laplace Transforms [[w]] Topological Notions [[w]] Measure-Theoretic Probability [[w]] Fourier Transforms [[w]] Filtrations and Optional Times [[w]] Martingales [[w]] Bibliography