SubjectsSubjects(version: 845)
Course, academic year 2018/2019
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Invariance Principles - NMTP434
Title in English: Principy invariance
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2019
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:4/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
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
Annotation -
Last update: T_KPMS (20.04.2015)
Probability measures on metric spaces. Prokhoroff theorem. Properties of C[0,1] and D[0,1]. Donsker invariance principle.
Aim of the course -
Last update: T_KPMS (16.05.2013)

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

Course completion requirements -
Last update: doc. RNDr. Petr Lachout, CSc. (11.10.2017)

The course is finalized by exam.

Literature - Czech
Last update: T_KPMS (20.04.2015)

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

Teaching methods -
Last update: T_KPMS (16.05.2013)

Lecture.

Requirements to the exam -
Last update: doc. RNDr. Petr Lachout, CSc. (11.10.2017)

The exam is oral.

Examination is checking knowledge of all matters read by the course lecturer.

Syllabus -
Last update: T_KPMS (20.04.2015)

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.

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

measure and integration theory, probability theory, functional analysis

 
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