**ELE525 Random Processes in Information Systems
**

Fall 1996-97

Professor Sergio Verdú

**Text:** G. R. Grimmett and D. R. Stirzaker, Probability and Random
Processes Second Edition, Oxford University, 1993

**1. Review of Basic Probability **

Probability Spaces - Independence Random - Variables Distribution Function - Random Vectors Conditional - Distribution Expectation and its Properties - Conditional Expectation - Characteristic function - Generating function

**2. Sequences of Random Variables **

Modes of Convergence - Relationships between convergence modes - Borel Cantelli Lemmas - Weak Law of Large Numbers - Strong Law of Large Numbers for independent sequences - Central Limit Theorem: Lyapunov and Lindeberg-Feller Theorems - Large Deviations for iid sequences - Martingales

**3. Random Processes **

Continuous and discrete-time processes - Stationarity - Wide-Sense Stationarity - Ergodicity - Mean Ergodic Theorem - Bikhoff-Khinchin Ergodic Theorem - Spectral Density of wide sense stationary processes - WSS processes and linear-time-invariant systems - Linear Minimum Mean-Square Estimation and Filtering - Karhunen-Loeve Expansion of Continuous-Time Processes

**4. Renewal Processes **

Memoryless property of exponential distribution - Poisson Processes - Nonhomogeneous Poisson Processes - Laws of large numbers for renewal processes; renewal theorem - Stopping Rules and Wald's identity - Renewal theory: steady-state distribution of age and residual life.

**5. Markov Processes **

(Discrete-Time) Markov Chains - Homogenous Markov chains - Irreducible chains - Positive Recurrent, Null Recurrent, Transient States and Mean recurrence time - Periodicity and ergodic states - Stationary distributions and limit theorems - Reversibility and Detailed Balance Equations - Continuous-Time Markov processes - Transition rate matrix and Holding times - Stationary distributions - Reversibility - Birth-Death Markov Processes - Applications to Queueing theory: exponential service queues

This page maintained by Michelle Young - Last Modified 11/20/96