Seminar: Convergence of random variables and probability metrics.

Seminar Supervisor

Prof. Dr. Evgeny Spodarev

Seminar Advisor

Abhinav Das

Date and place

The seminar will be in block (in two weeks on each Fridays). Four talks will held in one day.
Time: 8:00 hrs to 16:00 hrs.
Date: Will be updated Soon (Possibly in the last week of January.)
Venue : Will be updated soon.

Intended Audience

Bachelor and Master students in Mathematical programmes. 

Prerequisites

The level of difficulty in this seminar is varying between the different topics. The audience is at least supposed to be familiar with basic probability, statistics, basic analysis and measure theory. We ensure the participants that most of the 'beyond' knowledge will be learned during the seminar.

*Note: The language of instruction and communication will only be English

Contents

The development of the theory of probability metrics, a branch of probability theory, began with the study of problems related to limit theorems in probability theory. In general, the applicability of limit theorems stems from the fact that they can be viewed as an approximation to a given stochastic model and, consequently, can be accepted as an approximate substitute. The key question that arises in adopting the approximate model is the magnitude of the error that must be accepted. Because the theory of probability metrics studies the problem of measuring distances between random quantities or stochastic processes, it can be used to address the key question of how good the approximate substitute is for the stochastic model under consideration. Moreover, it provides the fundamental principles for building probability metrics – the means of measuring such distances.

In this seminar we focus on three specialized research direction in the theory of probability metric, as well as applications to different problems of probaility theory. The topics for the seminar is as follows:

  1. Introduction to Probability Distances and Probability Metrics - ch. 2, [1].
  2. Primary, Simple and Compound Probability Distances and Minimal and Maximal Distances and Norms - ch. 3, [1].
  3. Structural Classification of Probabilty Distances - ch. 4, [1].
  4. Monge–Kantorovich Mass Transference Problem, Minimal Distances and Minimal Norms - ch. 5, [1].
  5. Quantitative Relationships Between Minimal Distances and Minimal Norms - ch. 6, [1].
  6. Moment Problems Related to the Theory of Probability Metrics: Relations Between Compound and Primary Distances - ch. 9, [1].
  7. Introduction to Moment Distances . - ch. 10, [1].
  8. Uniformity in Weak and Vague Convergence  - ch. 11, [1].
  9. Glivenko–Cantelli Theorem and Bernstein–Kantorovich Invariance Principle - ch. 12, [1].
  10. Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements - ch. 15, [1]
  11. Statistical Estimates Obtained by the Minimal Distances Method - ch. 23, [1].
  12. Convergence of Random Variables and Distribution - ch. 5, [2].
  13. Sequence of Random Variables and Law of Large Numbers - ch. 4, [2].
  14. Relation between different notion of Convergence  [3]

Registration Details

Register yourself at Moodle page: Kurs: Stochastik (uni-ulm.de)

OR

To register for the seminar, please write an email to Abhinav Das until November, 6. In the e-mail please give your name, matriculation number, your programme of studies and subjects you have taken in the area of Probability or Statistics. Also, please indicate what chapter of the book you're interested in. 

Literature

  1. Svetlozar T. Rachev, Lev Klebanov, Stoyan V. Stoyanov, Frank Fabozzi - The Methods of Distances in the Theory of Probability and Statistics-Springer (2013)
  2. (Wiley series in probability and statistics. Probability and statistics section) Patrick Billingsley - Probability measures-Wiley (1986)
  3. Modes of Convergence, UC Berkeley (https://inst.eecs.berkeley.edu/~ee126/sp18/modes-of-convergence-short.pdf)

Contact

Seminar Supervisor

Prof. Dr. Evgeny Spodarev
Helmholtzstraße 18, Raum 1.65
Sprechzeiten: Nach Vereinbarung
E-Mail: Evgeny.Spodarev(at)uni-ulm.de
 

Seminar Advisor

Abhinav Das
Helmholtzstraße 18, Raum 1.40b
Sprechzeiten: Nach Vereinbarung
E-Mail: abhinav.das(at)uni-ulm.de