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Foundations of Stochastic Processes

From Probability Axioms to Random Matrix Theory

Rigorous graduate-level treatment of probability theory and stochastic processes for communications research: axioms, random variables, transforms, limit theorems, Markov chains, PSDs, and random matrix theory.

22 chapters194 sections65+ hours of content
Start from Chapter 1

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Foundations: Probability, Counting, and CombinatoricsRandom Variables and DistributionsMoment Methods and ConcentrationSpectral Analysis of Random ProcessesSpecial Classes of ProcessesAdvanced TopicsFuture topics

All Chapters

Part 1: Probability Foundations

Chapter 1

The Axioms of Probability

~160 minFoundational
Chapter 2

Conditional Probability and Independence

~160 minFoundational
Chapter 3

Reliability and Combinatorial Applications

~120 minFoundational
Chapter 4

Counting and Discrete Probability Models

~180 minIntermediate

Part 2: Random Variables and Distributions

Chapter 5

Discrete Random Variables

~180 minFoundational
Chapter 6

Continuous Random Variables

~240 minIntermediate
Chapter 7

Joint Distributions and Independence of Random Variables

~200 minFoundational
Chapter 8

Multivariate Distributions and the Gaussian

~240 minIntermediate

Part 3: Random Vectors, Transforms, and Limit Theorems

Chapter 9

Generating Functions and Transforms

~120 minIntermediate
Chapter 10

Probability Inequalities

~180 minIntermediate
Chapter 11

Convergence of Random Variables

~200 minIntermediate
Chapter 12

Conditional Expectation as a Random Variable

~150 minIntermediate

Part 4: Stochastic Processes: Fundamentals

Chapter 13

Introduction to Stochastic Processes

~150 minIntermediate
Chapter 14

Power Spectral Density

~150 minIntermediate
Chapter 15

Linear Systems with Random Inputs

~160 minIntermediate
Chapter 16

Mean-Square Calculus and Bandlimited Processes

~180 minAdvanced

Part 5: Markov Chains and Poisson Processes

Chapter 17

Discrete-Time Markov Chains

~220 minIntermediate
Chapter 18

Continuous-Time Markov Chains and Birth-Death Processes

~240 minIntermediate
Chapter 19

Gaussian Processes

~150 minIntermediate

Part 6: Advanced Topics

Chapter 20

Large Deviations and Concentration Inequalities

~150 minAdvanced
Chapter 21

Introduction to Random Matrix Theory

~180 minAdvanced
Chapter 22

Measure-Theoretic Foundations

~200 minAdvanced