Probabilistic processes

All lectures: All lectures at the Academy of Intelligence on the discipline: probabilistic processes. 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. Construction of a Sequence of Independent Real Random Variables with Given Distribution Functions
3. Random Walk as an Example of Random Processes
4. Finite-Dimensional Distributions of a Process. Kolmogorov's Theorems on Consistent Distributions. Consistency Conditions for Measures
5. Criterion for the Existence of a Process with Independent Increments. The Poisson and Wiener Processes
6. Gaussian processes. Construction of a real Gaussian process with given mean and covariance functions.
7. Construction of Brownian Motion via Schauder Functions
8. Undifferentiability (with probability 1) of the trajectories of Brownian motion at each point
9. Filtration. Markov Times, Stopping Time. Examples
10. Markov and Strong Markov Properties of Brownian Motion
11. Principle of reflection. Distribution Formulation of the law of the iterated logarithm
12. Weak convergence of probability measures. Theorem A.D. Alexandrova.
13. . 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)
14. Martingales, Submartingales, Supermartingales. Examples. Doob Decomposition.
15. . Дискретный вариант формулы Танака. Доказательство соотношения ,
16. Теорема Дуба о свободном выборе
17. . Неравенство Крамера-Лундберга.
18. Markov processes with discrete and continuous time. Examples
19. Proof that the actual process with independent increments is Markov
20. Построение марковской цепи по начальному распределению и переходным вероятностям. Пуассоновский процесс как цепь Маркова.
21. Monte Carlo Method: Essence and Examples of Application
22. Markov Transition Function. Homogeneous Markov Processes
23. Ergodic Theorem for Continuous-Time Markov Chains
24. . Stationary distribution. Erlang Formulas (model description).
25. Kolmogorov's Differential Equations (Direct and Backward)
26. The integral over an orthogonal random measure (in the case of a finite and -finite structure measure).
27. The Karhunen-Loève Theorem
28. Herglotz theorem. Formulation of the Bochner-Khinchin Theorem
29. Stationary in a broad sense processes, their spectral representation. Ergodicity in L2 ().
30. Spectral density. The moving-average process as a process possessing a spectral density.
31. Regular and singular processes. Formulation of the Wold theorem and Kolmogorov theorem (a regularity criterion in terms of spectral density).
32. The Langevin Equation. The Ornstein-Uhlenbeck Process
33. The Stochastic Integral: The Ito, Wiener and Stratonovich Integrals and Their Properties
34. Stochastic Differential Equations and Strong Solutions
35. Ito's Formula and an Example of Its Use
36. Random process
37. Markov process
38. Markov chain
39. Non-Markov process
40. The Cramer-Lundberg Model as an Example of a Random Process
41. Poisson Random Measures
42. Empirical Measure as a Random Process
43. The Renewal Process as a Random Process - 2.1. Definition of the Renewal Process
44. 2.2. The Renewal Function and Its Properties
45. 2.3. Integral Renewal Equations
46. 2.4. Renewal Density
47. 2.5. Asymptotic Behavior of the Renewal Function (the Elementary Renewal Theorem)
48. 2.6. Terminating Renewal Processes
49. 2.7. Key Renewal Theorem
50. 2.8 Characteristics of random variables associated with the renewal process
51. 2.9. Examples of using the key renewal theorem
52. 2.10. Some Useful Estimates for Renewal Functions
53. 2.11. Stationary Renewal Processes
54. 2.12. Alternating Renewal Processes
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