Course Program

Published

8 September 2026

The course is built as a direct continuation of ETMT: overlaps with it are compressed to brief reminders, and the freed-up time is devoted to topics that are absent from self-contained probability theory courses or are given only in passing (uniform integrability, the Borel-Cantelli lemma and the zero-one law as a separate topic, the Lindeberg-Lyapunov conditions, conditional distributions with a Gaussian forecast). The throughline is computational technique for the continuous case: Jacobians, convolutions, and gamma- and beta-integrals.

Part 1. Probability on the Foundation of Measure. Convergences

  1. The language of probability theory (1-2 seminars, β€œpaying off debts” from ETMT). Kolmogorov’s axioms; independence and the criterion via \(\pi\)-systems; random variables and distributions; expectation as a Lebesgue integral, change of variables, generalized densities.
  2. Random vectors. Joint distributions, multivariate distribution functions; the density transformation formula (Jacobian); marginals; independence criteria; the main continuous distributions in practice.
  3. Convolution. Covariance and inequalities. The convolution formula as a technique; covariance and correlation in the general case, the covariance matrix; Jensen’s, Lyapunov’s, and Cauchy-Bunyakovsky-Schwarz inequalities.
  4. Convergence of random variables. The complete diagram of the four types of convergence with counterexamples; the criterion for almost sure convergence; inheritance of convergence, Slutsky’s theorem; uniform integrability and convergence of moments.
  5. The Borel-Cantelli lemma and the zero-one law. Both parts of the lemma; problems on \(\limsup\) of events; tail \(\sigma\)-algebras, Kolmogorov’s zero-one law; optional β€” the three-series theorem.
  6. Laws of large numbers. The Cantelli strong law of large numbers; Kolmogorov’s inequality, the Kolmogorov-Khinchin theorem; Toeplitz’s and Kronecker’s lemmas; Kolmogorov’s strong law of large numbers; applications: the Monte Carlo method, normal numbers.

Midterm exam 1.

Part 2. Limit Theorems and Conditional Constructions

  1. Weak convergence of probability measures. Convergence in the main sense; Alexandrov’s theorem; tightness of families of measures; Prokhorov’s theorem.
  2. Characteristic functions. Properties, computational technique; moments and derivatives; the uniqueness theorem, the inversion formula; the Bochner-Khinchin theorem.
  3. The central limit theorem. The continuity theorem; proof of the CLT; the Lindeberg and Lyapunov conditions; the Berry-Esseen inequality; the de Moivre-Laplace theorems, the Poisson approximation; optional β€” the law of the iterated logarithm (statement).
  4. Gaussian vectors. Three equivalent definitions; independence \(\Leftrightarrow\) uncorrelatedness; density, linear transformations; the multivariate CLT.
  5. Conditional expectation. The discrete case; definition via a \(\sigma\)-algebra; the complete set of properties (the foundation for martingales in the stochastic processes course).
  6. Conditional distributions and computing conditional expectation. Conditional densities, \(\mathbb{E}[\xi \mid \eta = y]\); the best forecast in \(L^2\); the Gaussian conditional expectation as a linear forecast β€” a bridge to regression in statistics and to Gaussian processes.

Midterm exam 2.