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Course, academic year 2023/2024
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Invariance Principles - NMTP434
Title: Principy invariance
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:4/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Petr Lachout, CSc.
Class: M Mgr. PMSE
M Mgr. PMSE > Povinně volitelné
Classification: Mathematics > Probability and Statistics
Pre-requisite : NMSA405
Is interchangeable with: NSTP125
Annotation -
Probability measures on metric spaces. Prokhoroff theorem. Properties of C[0,1] and D[0,1]. Donsker invariance principle.
Last update: T_KPMS (20.04.2015)
Aim of the course -

To teach and explain theory of convergence of random processes, especially in functional spaces C([0,1]) and D([0,1]).

Last update: T_KPMS (16.05.2013)
Course completion requirements -

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Course finalization

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The course is finalized by exam.

Last update: Lachout Petr, doc. RNDr., CSc. (29.04.2020)
Literature - Czech

Billingsley, P.: Convergence of Probability Measures, John Wiley & Sons,New York, 1968.

Čech, E.: Topologické prostory, Academia, Praha, 1959.

Kelley, J.L.: General Topology, D. van Nostrand Comp., New York, 1955.

Štěpán J.: Teorie pravděpodobnosti. Matematické základy. Academia, Praha 1987

Last update: T_KPMS (20.04.2015)
Teaching methods -

Lecture.

Last update: T_KPMS (16.05.2013)
Requirements to the exam -

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Requirements to exam

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The exam is oral.

Examination is checking knowledge of all topics read at the lecture and parts given to self-study by the course lecturer.

Last update: Lachout Petr, doc. RNDr., CSc. (14.02.2024)
Syllabus -

1. Basic of topology (product and relativ topology, Tikhonov theorem, random maps, random variables, probability measures on topological spaces, weak convergence of probability measures).

2. Metric spaces (Polish space, Prokhorov theorem, Banach space).

3. Topology of the space of functions (Borel sigma-algebra, Daniell-Kolmogorov theorem, cylindric sigma-algebra, random process).

4. Properties of spaces C[0,1] and D[0,1],

5. Donsker invariance princip and applications.

Last update: T_KPMS (20.04.2015)
Entry requirements -

measure and integration theory, probability theory, functional analysis

Last update: Lachout Petr, doc. RNDr., CSc. (30.05.2018)
 
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