Ability to create and analyze stochastic models ,
deepening in the field of mathematical probability
multivariate limit distribution rate, symmetric distributions,
Zero-one law, dependences of multivariate distributions and characterisations
characteristic functions, Levy'scher continuity theorem, three-series theorem,
infinitely divisible distributions and Levy-Khinchin formula, stable distributions,
Poisson point process
convergencies of measures
vague convergence, portmanteau theorem, theorem of Slutsky,
Tightness, Theorem of Prohorov, select theorem of Helly,
Product spaces (infinite dimensional) projective family, semi-Markov kernels and groups, Kolmogorov' extension
Interchangeability, theorem of Definetti,
Markov chains, invariant measures, convergence theorem for Markov chains, Monte Carlo method ,ergodic theorems
mixing systems,
Martingales, stopping times, discrete stochastic integral Martingaldarstellungssatz, Doob decomposition,
optional sampling and optional stopping, Martingalkonvergenzsätze,
Rückwärtsmartingale
Laws of the iterated logarithm, Skorohod'sche embedding theorem, theorem of Hartman and Wintner,
Books (recommended)
H. Bauer - Wahrscheinlichkeitstheorie - de Gruyter
P. Billingsley - Probability and Measure - Wiley
P. Billingsley - Convergence of Probability Measures - Wiley
R. M. Dudley - Real Analysis and Probability - Cambridge University Press
B. Gnedenko - Theory of Probability - Gordon and Breach Science Publishers/AMS Chelsey Pub.
S. Karlin & H. Taylor - A first course in stochastic processes - Academic Press
S. Karlin & H. Taylor - A second course in stochastic processes - Academic Press
A. Klenke - Wahrscheinlichkeitstheorie - Springer
N. Kusolitsch - Maß- und Wahrscheinlichkeitstheorie - Springer
S. Ross - Introduction to. Probability Models - Academic Press
A. Shiryaev Probability - Springer
K. Schürger - Wahrscheinlichkeitstheorie - Oldenbourg
