# Topics - Conditional expectation: Discrete case, Gaussian case, conditional density functions; existence and uniqueness; basic properties. - Martingales: Martingales and submartingales in discrete time; optional stopping; Doob’s inequalities, upcrossings, martingale convergence theorems; applications of martingale techniques. - Stochastic processes in continuous time: Kolmogorov’s criterion, regularization of paths; martingales in continuous time. - Weak convergence: Definitions and characterizations; convergence in distribution, tightness, Prokhorov’s theorem; characteristic functions, Lévy’s continuity theorem. - Sums of independent random variables: Strong laws of large numbers; central limit theorem; Cramér’s theory of large deviations. - Brownian motion: Wiener’s existence theorem, scaling and symmetry properties; martingales associated with Brownian motion, the strong Markov property, hitting times; properties of sample paths, recurrence and transience; Brownian motion and the Dirichlet problem; Donsker’s invariance principle. - Poisson random measures: Construction and properties; integrals. - Lévy processes: Lévy-Khinchin theorem.