Institute of Stochastics

Stable Distributions

 Lecturer
Prof. Dr. Evgeny Spodarev

Teaching Assistant
Dr. Jürgen Kampf

Time and Place

Lecture
Thursday, 14-16, Heho 18, room E.60

Exercise Session
Tuesday, 14-16, Heho 18, room E.60

Type

2 hours lecture + 1 hours exercises

Prerequisites

Basic analysis and linear algebra courses, basic probability course (Elementare WR und Statistik).

 

Intended audience

Students of master of mathematics, master of business mathematics, master of mathematical biometry, master of finance.

 

Content

In modern applications, there is a need to model phenomena that can be measured by very high numerical values which occur rarely. In probability theory, one talks about distributions with heavy tails. One class of such distributions are stable laws which (apart from the Gaussian one) do not have a finite variance. They possess a number of striking properties which make them inevitable in modelling of processes in radioelectronics, engineering, radiophysics, astrophysics and cosmology, finance, insurance, etc., to name just a few. This introductory lecture is devoted to basic properties of such distributions.

Main topics are

1) Stability with respect to convolution
2) Characteristic functions and densities
3) Non-Gaussian limit theorem for i.i.d. random summands
4) Representations and tail properties, symmetry and skewness
5) Simulation

Exam

Oral exam

Prerequisite: 50% of all credits from the exercise sheet.

Exercise sheets

Exercise sheets can be only found in moodle.

 Lecture notes

 Lecture notes can be downloaded here.

Literature

  • J. Nolan. Stable Distributions – Models for Heavy Tailed Data. Birkhäuser, Boston, 2013.
  • G. Samorodnitsky, M.S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall, New York, 1994.
  • K.-I. Sato. Lévy Processes and Infinite Divisibility. Cambridge University Press, Cambridge, 1999 (Chapter 3).
  • V. M. Zolotarev. One-Dimensional Stable Distributions. Translations of Mathematical Monographs, vol 65, AMS, Providence RI, 1986.
  • S. T. Rachev, S. Mittnik. Stable Paretian Models in Finance. Wiley, New York, 2000.
  • V.V. Uchaikin, V. M. Zolotarev. Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht, 1999.

Link to the course reserve of the library.