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
  1. What is probability
    • Introduction
    • Some paradoxons in probability theory
    • Some applications of probability theory
  2. 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
  3. 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
  4. 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
  5. 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

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