Lecturer
Prof. Dr. Evgeny Spodarev

Teaching Assistant
Dr. Vitalii Makogin

---

Time and Place

Lecture
Tuesday, 10-12, Heho 18, room E60

Exercise Session
Thursday, 12-13, Heho 18, room E60

---

Type

2 hours lecture + 2 hours exercises (each 15 days)

---

Prerequisites

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

---

Intended Audience

Students of Bachelor Program in Mathematics, Mathematics and Management, Education.

---

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

Additional topics may be

6) Multivariate stable distributions
7) Elementary stable random processes

---

Criteria to pass the final exam

50 % of credits and sufficient evaluation of the written exam.

Scoring: use SLC login.

Dates:

Final exam: tba
Second exam: tba

---

Exercise sheets

• Exercise sheet 1
• Exercise sheet 2, till 3d of May
• Exercise sheet 3
• Exercise sheet 4
• Exercise sheet 5
• Exercise sheet 6
• Exercise sheet 7

  

 Lecture notes

 The lecture notes for stabla distributions can be found 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.