M345P6 (Advanced) Probability Theory

Lectures Spring 2020

Mo 13:00 – 15:00/ HXLY 144 / & Thu 10:00 - 11:00 / HXLY 213 Clore ( First week only HXLY 139 ) /

Office Hours (Room: 6M55 )

M345P6 Tue 12:00-13:00

Syllabus

A rigorous approach to the fundamental properties of probability:

Probability measures. Random variables. Independence. Sums of independent random variables; Weak and Strong Laws of Large Numbers. Weak convergence, characteristic functions, Central Limit Theorem. Elements of Brownian motion. Ergodic Theory.



Proba Content 2020



Problem Sets

PS.1 ; PS.2 ; PS.3 ; PS.4 ; PS.5 ;

PS.1slns ; PS.2slns ; PS.3slns ; PS.4slns ; PS.5slns ;

Course Work

1.CW : 24 January

2.CW : 21 February

3.CW : 13 March

Mastery Material 2020



Notes

PROBA2019_Notes

PROBA2020_Notes

Content 2020 :

PROBA2020-Part.1

Notes on Conditional Expectations

CURRENT READING

I.

II.

III.

IV.



BIBLIOGRAPHY

Leonid Koralov, Yakov G. Sinai, Theory of Probability and Random Processes(+pdf)

Sinai, Y.G., Probability theory : an introductory course (Google Books)

Imperial College Library Electronic Resources



Kolmogorov, A. N. Foundations of the theory of probability (+pdf)

Kac, Mark, Statistical Independence in Probability, Analysis and Number Theory.

Patrick Billingsley Probability and Measure (pdf)

Stroock, Daniel W., Probability Theory: An analytic view (Ch. I)

Schilling, Rene L., Measures, integrals and martingales

Stroock, Daniel W., Mathematics of Probability

Feller, William , An Introduction to Probability Theory and Its Applications,

Williams, D., Probability with Martingales.

Stroock, Daniel W., A concise introduction to the theory of integration








Some LINKS


History of Mathematics :

A Simple Pole in Ithaca, NY” by Daniel W. Stroock

Mark Kac at St-Andrews Math History

Review of Enigma of Chance

Théorie analytique des probabilités; by Laplace, Pierre Simon, marquis de, 1749-1827

(http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAPLACE__7_R2_0)

Özlem Kart, A Historical Survey of the Development of Classical Probability Theory


Glenn Shafer & Vladimir Vovk The origins and legacy of Kolmogorov’s Grundbegriffe

Mathematicians:

Mikhail Lifshits, Lectures on Gaussian Processes

Roland Speicher, Free Probability Theory, And Its Avatars in Representation Theory, Random Matrices,and Operator Algebras; also Featuring: Non-commutative Distributions






Math Info

Vitali nonmeasurable set

M.G. Nadkarni and V.S. Sunder
Hamel bases and measurability

Gaussian Random Variables

Gibbs Random Fields

Fundamentals of Stein’s method∗Nathan Ross




Mid-term lecture feedback questions