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Probabilistic processes

All lectures: All lectures on the discipline: probabilistic processes are collected here. These fascinating lectures provide us with the opportunity to explore the deep principles and phenomena underlying our knowledge and our desire to understand the world.


1. Random elements and their distribution. A random process as a family of random elements and as one measurable mapping.
2. . Undifferentiability (with probability 1) of the trajectories of Brownian motion at each point.
3. Principle of reflection. Distribution Formulation of the law of the iterated logarithm
4. Weak convergence of probability measures. Theorem A.D. Alexandrova.
5. . Preservation of weak convergence under the action of continuous mappings. Formulation of the Prokhorov theorem on the density of a family of measures. The principle of invariance (formulations of theorems of Donsker, Prokhorov, Borovkov, Skorokhod)
6. . Дискретный вариант формулы Танака. Доказательство соотношения ,
7. Теорема Дуба о свободном выборе
8. . Неравенство Крамера-Лундберга.
9. Markov processes with discrete and continuous time. Examples
10. Proof that the actual process with independent increments is Markov
11. Построение марковской цепи по начальному распределению и переходным вероятностям. Пуассоновский процесс как цепь Маркова.
12. . Stationary distribution. Erlang Formulas (model description).
13. The integral over an orthogonal random measure (in the case of a finite and -finite structure measure).
14. Herglotz theorem. Formulation of the Bochner-Khinchin Theorem
15. Stationary in a broad sense processes, their spectral representation. Ergodicity in L2 ().
16. Regular and singular processes. Formulation of the Wold theorem and Kolmogorov theorem (a regularity criterion in terms of spectral density).
17. Random process
18. Markov process
19. Markov chain
20. Non-Markov process
Complain