. 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)

Lecture



. 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). Formulation of the Strassen theorem.

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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.
Construction of a Sequence of Independent Real Random Variables with Given Distribution Functions
Random Walk as an Example of Random Processes
Finite-Dimensional Distributions of a Process. Kolmogorov's Theorems on Consistent Distributions. Consistency Conditions for Measures
Criterion for the Existence of a Process with Independent Increments. The Poisson and Wiener Processes
Gaussian processes. Construction of a real Gaussian process with given mean and covariance functions.
Construction of Brownian Motion via Schauder Functions
Undifferentiability (with probability 1) of the trajectories of Brownian motion at each point
Filtration. Markov Times, Stopping Time. Examples
Markov and Strong Markov Properties of Brownian Motion
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)
Martingales, Submartingales, Supermartingales. Examples. Doob Decomposition.
. 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.
Monte Carlo Method: Essence and Examples of Application
Markov Transition Function. Homogeneous Markov Processes
Ergodic Theorem for Continuous-Time Markov Chains
. Stationary distribution. Erlang Formulas (model description).
Kolmogorov's Differential Equations (Direct and Backward)
The integral over an orthogonal random measure (in the case of a finite and -finite structure measure).
The Karhunen-Loève Theorem
Herglotz theorem. Formulation of the Bochner-Khinchin Theorem
Stationary in a broad sense processes, their spectral representation. Ergodicity in L2 ().
Spectral density. The moving-average process as a process possessing a spectral density.
Regular and singular processes. Formulation of the Wold theorem and Kolmogorov theorem (a regularity criterion in terms of spectral density).
The Langevin Equation. The Ornstein-Uhlenbeck Process
The Stochastic Integral: The Ito, Wiener and Stratonovich Integrals and Their Properties
Stochastic Differential Equations and Strong Solutions
Ito's Formula and an Example of Its Use
Random process
Markov process
Markov chain
Non-Markov process
The Cramer-Lundberg Model as an Example of a Random Process
Poisson Random Measures
Empirical Measure as a Random Process
The Renewal Process as a Random Process - 2.1. Definition of the Renewal Process
2.2. The Renewal Function and Its Properties
2.3. Integral Renewal Equations
2.4. Renewal Density
2.5. Asymptotic Behavior of the Renewal Function (the Elementary Renewal Theorem)
2.6. Terminating Renewal Processes
2.7. Key Renewal Theorem
2.8 Characteristics of random variables associated with the renewal process
2.9. Examples of using the key renewal theorem
2.10. Some Useful Estimates for Renewal Functions
2.11. Stationary Renewal Processes
2.12. Alternating Renewal Processes