MA 144 a/b Probability

 This is an introduction into the theory of probability This course was given in in the first and second term 1995 at Caltech. Here is the PS file 1.4 MBytes, 277 pages of the course notes. Content What is probability Introduction Some paradoxons in probability theory Some applications of probability theory Limit theorems Probability spaces, random variables, independence Kolmogorov's 0-1 law, the Borel-Cantelli lemma Integration, Expectation, Variance Some inequalities The weak law of large numbers Convergence of random variables The strong law of large numbers Birkhoff's ergodic theorem Kolmogorov's inequlity, three series theorem, Levy's theorem Distribution functions The central limit theorem Entropy of distributions Gibbs distributions Markov operators Characteristic functions The law of the iterated logarithm Use of characteristic funcdtions Discrete martingales Conditional expectation Martingales Stopping times Doob's convergence theorem Computation of a limiting density Extinction probability for the branching process Levy's upward and downward theorems Doob's decomposition of a stochastic process Doob's submartingal inequality Doob's L p inequality Random walks The arc-sin law for the 1D random walk Random walk on a free group Distribution of the first return time The free laplacian on a discrete group The discrete Feynmann-Kac formula Markov chains Stochastic calculus Brownian motion History of Brownian motion Overview over other existence proofs Properties of Brownian motion Other Brownian processes The Wiener measure Levy's modulus of continuity Stopping times Relation with potential theory Martingales Doob inequality Kinthcine's law of iterated logarithm Theorem of Dynkin-Hunt Selfintersection of Brownian motion Recurrence of Brownian motion Feynman-Kac for the oscillator Wiener sausage The Ito integral for Brownian motion Ito's formula Processes of bounded quadratic variation The Ito integral for martingales Stochastic differential equations Selected topics Percolation FKG correlation inequality Russo's formula Mean size of the open cluster The average number of open clusters Localisation of random Jacobi matrices