Random Processes

I. Introduction.- II. Basic Notions for Finite and Denumerable State Models.- a. Events and Probabilities of Events.- b. Conditional Probability, Independence, and Random Variables.- c. The Binomial and Poisson Distributions.- d. Expectation and Variance of Random Variables (Moments).- e. The Weak Law of Large Numbers and the Central Limit Theorem.- f. Entropy of an Experiment.- g. Problems.- III. Markov Chains.- a. The Markov Assumption.- b. Matrices with Non-negative Elements (Approach of Perron-Frobenius).- c. Limit Properties for Markov Chains.- d. Functions of a Markov Chain.- e. Problems.- IV. Probability Spaces with an Infinite Number of Sample Points.- a. Discussion of Basic Concepts.- b. Distribution Functions and Their Transforms.- c. Derivatives of Measures and Conditional Probabilities.- d. Random Processes.- e. Problems.- V. Stationary Processes.- a. Definition.- b. The Ergodic Theorem and Stationary Processes.- c. Convergence of Conditional Probabilities.- d. MacMillan's Theorem.- e. Problems.- VI. Markov Processes.- a. Definition.- b. Jump Processes with Continuous Time.- c. Diffusion Processes.- d. A Refined Model of Brownian Motion.- e. Pathological Jump Processes.- f. Problems.- VII. Weakly Stationary Processes and Random Harmonic Analysis.- a. Definition.- b. Harmonic Representation of a Stationary Process and Random Integrals.- c. The Linear Prediction Problem and Autoregressive Schemes.- d. Spectral Estimates for Normal Processes.- e. Problems.- VIII. Martingales.- a. Definition and Illustrations.- b. Optional Sampling and a Martingale Convergence Theorem.- c. A Central Limit Theorem for Martingale Differences.- d. Problems.- IX. Additional Topics.- a. A Zero-One Law.- b. Markov Chains and Independent Random Variables.- c. A Representation for a Class of Random Processes.- d. A Uniform Mixing Condition and Narrow Band-Pass Filtering.- e. Problems.- References.