Probability Theory II

  1. Martingale theory
    2. Optional sampling; optional stopping; Doob's inequality; martingale convergence theorem; backward martingales; de Finetti's theorem
  2. Convergence of measures
    3. Weak and vague convergence; Prohorov's theorem
  3. Product measures
    4. Theorem of Ionescu-Tulcea; consistent families; Kolmogorov's extension theorem
  4. Central limit theorem
    5. Theorem of Stone-Weierstrass; characteristic functions; the central limit theorem
  5. Brownian motion
    6. Stochastic processes; Gauss processes; construction and path properties of Brownian motion; strong Markov property and reflection principle; law of the iterated logarithm; Skorohod's embedding theorem
  6. Poisson point processes