Regular and singular processes. Formulation of the Wold theorem and Kolmogorov theorem (a regularity criterion in terms of spectral density).

Lecture



Regular and singular processes. Formulation of the Wold theorem and Kolmogorov theorem (a regularity criterion in terms of spectral density).

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Lectures and tutorial on "probabilistic processes"

Terms: probabilistic processes

Random elements and their distribution. A random process as a family of random elements and as one measurable mapping.
. Undifferentiability (with probability 1) of the trajectories of Brownian motion at each point.
Principle of reflection. Distribution Formulation of the law of the iterated logarithm
Weak convergence of probability measures. Theorem A.D. Alexandrova.
. 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)
. Дискретный вариант формулы Танака. Доказательство соотношения ,
Теорема Дуба о свободном выборе
. Неравенство Крамера-Лундберга.
Markov processes with discrete and continuous time. Examples
Proof that the actual process with independent increments is Markov
Построение марковской цепи по начальному распределению и переходным вероятностям. Пуассоновский процесс как цепь Маркова.
. Stationary distribution. Erlang Formulas (model description).
The integral over an orthogonal random measure (in the case of a finite and -finite structure measure).
Herglotz theorem. Formulation of the Bochner-Khinchin Theorem
Stationary in a broad sense processes, their spectral representation. Ergodicity in L2 ().
Regular and singular processes. Formulation of the Wold theorem and Kolmogorov theorem (a regularity criterion in terms of spectral density).
Random process
Markov process
Markov chain
Non-Markov process