## MATH 228: Stochastic Methods in Engineering (CME 308, MS&E 324)

The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and an introduction to stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.

Terms: Spr
| Units: 3

Instructors:
Glynn, P. (PI)

## MATH 228A: Probability, Stochastic Analysis and Applications

The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and basic stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.

Last offered: Spring 2016

## MATH 230A: Theory of Probability I (STATS 310A)

Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites:
STATS 116,
MATH 171.

Terms: Aut
| Units: 3

## MATH 230B: Theory of Probability II (STATS 310B)

Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,n(v) ergodic theory. Prerequisite: 310A or
MATH 230A.

Terms: Win
| Units: 3

Instructors:
Dembo, A. (PI)

## MATH 230C: Theory of Probability III (STATS 310C)

Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or
MATH 230B.
http://statweb.stanford.edu/~adembo/stat-310c/

Terms: Spr
| Units: 3

Instructors:
Montanari, A. (PI)

## MATH 231: Mathematics and Statistics of Gambling (STATS 334)

Probability and statistics are founded on the study of games of chance. Nowadays, gambling (in casinos, sports and the Internet) is a huge business. This course addresses practical and theoretical aspects. Topics covered: mathematics of basic random phenomena (physics of coin tossing and roulette, analysis of various methods of shuffling cards), odds in popular games, card counting, optimal tournament play, practical problems of random number generation. Prerequisites: Statistics 116 and 200.

Terms: Aut
| Units: 3

## MATH 232: Topics in Probability: Percolation Theory

An introduction to first passage percolation and related general tools and models. Topics include early results on shape theorems and fluctuations, more modern development using hyper-contractivity, recent breakthrough regarding scaling exponents, and providing exposure to some fundamental long-standing open problems. Course prerequisite: graduate-level probability.

Last offered: Autumn 2016
| Repeatable
for credit

## MATH 233A: Topics in Combinatorics

A topics course in combinatorics and related areas. The topic will be announced by the instructor.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Last offered: Autumn 2019
| Repeatable
for credit

## MATH 233B: Topics in Combinatorics

A topics course in combinatorics and related areas. The topic will be announced by the instructor.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Terms: Win
| Units: 3
| Repeatable
for credit

Instructors:
Huh, J. (PI)

## MATH 233C: Topics in Combinatorics

A topics course in combinatorics and related areas. The topic will be announced by the instructor.

Terms: Spr
| Units: 3
| Repeatable
for credit

Instructors:
Fox, J. (PI)

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