Probability Theory 1
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Discrete mathematics1L (mm1c1vm1)
Syllabus designed by:
Introduction of the basic notions and results of discrete probability theory without measure theoretic foundations.
- W. Feller: An Introduction to Probability Theory and Its Applications. Vol. 1, 3rd edition, Wiley, 1968.
- Probability space, conditional probablity, independence. Poincaré’s and Jordan’s formulae, the method of Rényi. Random variables, expectation, variance, covariance, correlation. Generating function.
- Simple symmetric random walk. Reflection principle and its applications. Distribution of random variables connected with the random walk: Position, times of recurrence, last visits and long leads, arcsine law, changes of sine, maxima and first passages, number of recurrences.
- Markov’s and Chebyshev’s inequalities, the weak law of large numbers. Convergence in probability. Moment generating function, rate function. Exponential inequalities (Chernoff-Hoeffding, Azuma).
- Asymptotic results for the binomial distribution. The DeMoivre-Laplace limit theorems, local and global versions. Limit distributions of random variables connected with the random walk. Poisson approximation via coupling, LeCam’s theorem.
- Almost sure convergence. The Borel-Cantelli lemma, the strong law of large numbers, Borel’s theorem. The law of iterated logarithm for symmetric Bernoulli variables. Lévy classes, EFKP theorem.
- Generating functions, connection with weak convergence. Random sums. Characterization of discrete infinitely divisible distributions. Point processes with independent and stationary increments. Branching processes, the probability of extinction.