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Topics in probability and statistics

2024/2025
Programme:
Doctoral Programme Mathematics and Physics
Orientation:
Mathematics
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
ECTS:
6
Language:
slovenian, english
Course director:

Dr. Janez Bernik, Assoc. Prof. Dr. Mihael Perman

Lecturer (contact person):

Assist. Prof. Dr. Matija Vidmar

Hours per week – 1. or 2. semester:
Lectures
2
Seminar
0
Tutorial
0
Lab
0
Prerequisites

There are no prerequisites.

Content (Syllabus outline)

The content consists of a selection of standard topics in advanced probability and statistics. Possible topics are:

  • Convergence of ramdom variables.
  • Conditional expectation and conditional distributions.
  • Martingales.
  • Brownian motion.
  • Parameter estimation, optimality.
  • Hypothesis testing, optimality.
  • Asymptotic theory
    The choice depends on students' research interests.
Readings
  1. J. Bertoin: Lévy processes, Cambridge : Cambridge University Press, cop. 1996.
  2. R. Durrett: Probability : theory and examples, 2nd ed., Belmont (Calif.) : Duxbury Press : Wadsworth : ITP International Thompson Publ., cop. 1996.
  3. S. M. Ross: Stochastic processes, 2nd ed., New York : Wiley, cop. 1996.
  4. M. J. Schervish: Theory of statistics, 2nd corrected print. - New York : Springer, cop. 1997.
  5. J. Shao: Mathematical statistics, 2nd ed. - New York : Springer, cop. 2010.
  6. S. R. S. Varadhan: Probability theory, New York : Courant Institute of Mathematical Sciences ; Providence (Rhode Island) : American Mathematical Society, cop. 2001.
Objectives and competences

The main goal of the course is to provide students with some important topics in probability and statistics.

Intended learning outcomes

Znanje in razumevanje predstavljenih konceptov.
Sposobnost uporabe pridobljenega znanja in spretnosti.

Learning and teaching methods

Lectures, consultations, problem sessions

Assessment

Writen exam (homeworks), oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Realizing irreducible semigroups and real algebras of compact operators. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2008, vol. 348, no. 2, str. 692-707. [COBISS-SI-ID 14899289]
BERNIK, Janez, MARCOUX, Laurent W., RADJAVI, Heydar. Spectral conditions and band reducibility of operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 86, no. 1, str. 214-234. [COBISS-SI-ID 16357721]
BERNIK, Janez, MASTNAK, Mitja. Lie algebras acting semitransitively. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2013, vol. 438, iss. 6, str. 2777-2792. [COBISS-SI-ID 16553561]
PERMAN, Mihael. An excursion approach to Ray-Knight theorems for perturbed Brownian motion. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 1996, let. 63, str. 67-74. [COBISS-SI-ID 7621465]
PERMAN, Mihael, WELLNER, Jon A. On the distribution of Brownian areas. Annals of applied probability, ISSN 1050-5164, 1996, let. 6, št. 4, str. 1091-1111. [COBISS-SI-ID 7101017]
PERMAN, Mihael, WELLNER, Jon A. An excursion approach to maxima of the Brownian bridge. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 2014, vol. 124, iss. 9, str. 3106-3120. [COBISS-SI-ID 17154393]