Probability and Statistics

0 Introduction to Random Processes.- 0.1. Random Evolution Through Time.- 0.2. Basic Measure Theory.- 0.3. Convergence in Distribution.- 1 Time Series.- 1.1. Second Order Processes.- 1.2. Spatial Processes with Orthogonal Increments.- 1.3. Stationary Second Order Processes.- 1.4. Time Series Statistics.- 2 Martingales in Discrete Time.- 2.1. Some Examples.- 2.2. Martingales.- 2.3. Stopping.- 2.4. Convergence of a Submartingale.- 2.5. Likelihoods.- 2.6. Square Intergrable Martingales.- 2.7. Almost Sure Asymptotic Properties.- 2.8. Central Limit Theorems.- 3 Asymptotic Statistics.- 3.1. Models Dominated at Each Instant.- 3.2. Contrasts.- 3.3. Rate of Convergence of an Estimator.- 3.4. Asymptotic Properties of Tests.- 4 Markov Chains.- 4.1. Introduction and First Tools.- 4.2. Recurrent or Transient States.- 4.3. The Study of a Markov Chain Having a Recurrent State.- 4.4. Statistics of Markov Chains.- 5 Step by Step Decisions.- 5.1. Optimal Stopping.- 5.2. Control of Markov Chains.- 5.3. Sequential Statistics.- 5.4. Large Deviations and Likelihood Tests.- 6 Counting Processes.- 6.1. Renewal Processes and Random Walks.- 6.2. Counting Processes.- 6.3. Poisson Processes.- 6.4. Statistics of Counting Processes.- 7 Processes in Continuous Time.- 7.1. Stopping Times.- 7.2. Martingales in Continuous Time.- 7.3. Processes with Continuous Trajectories.- 7.4. Functional Central Limit Theorems.- 8 Stochastic Integrals.- 8.1. Stochastic Integral with Respect to a Square Integrable Martingale.- 8.2. Ito's Formula and Stochastic Calculus.- 8.3. Asymptotic Study of Point Processes.- 8.4. Brownian Motion.- 8.5. Regression and Diffusions.- Notations and Conventions.