# Geometric methods in probability theory

*Group leader: Prof. Dr. Ilya Molchanov*

Geometric probabilities are as old as probability theory itself. Modern probability theory deals with random objects in general spaces, some of them bearing clear geometric features. Such objects are random convex sets, random measures, or random metric spaces. Apart from mathematical studies of such objects, the probability group in Bern focuses on geometric aspects which arise in the studies of very classical probabilistic models, like multivariate probability distributions, most importantly stable distributions. A particular attention is paid to applications of random sets in mathematical finance and economics.

The probability group in Bern collaborates with researchers from the University of Zagreb, Karlsruhe Institute of Technology, Kiev State University, University of Götheborg, Universidad Carlos III de Madrid, and Cornell University, among others.

#### Selected publications

The main theme of research deals with random sets and point processes. Numerous results in this area are presented in the thoroughly revised and extended second edition of Ilya Molchanov’s book on random sets.

- I. Molchanov. Theory of Random Sets. 2nd edition, Springer, London, 2017. [SpringerLink]

Further recent papers representing various aspects of research activities are:

- A. Marynych and I. Molchanov. Facial structure of strongly convex sets generated by random samples. Advances in Mathematics, 2021. [DOI:10.1016/j.aim.2021.108086]
- I. Cascos, Q. Li. and I. Molchanov. Depth and outliers for samples of sets and random sets distributions. Australian and New Zealand Journal of Statistics, 2021, 63, 55-82. doi: [DOI:10.1111/anzs.12326]
- I. Molchanov and A. Mühlemann. Nonlinear expectations of random sets. Finance and Stochastics, 2021, 25, 5-41. [DOI:10.1007/s00780-020-00442-3]
- I. Molchanov and R. Turin. Convex bodies generated by sublinear expectations of random vectors. Advances in Applied Mathematics, 2021, Paper No, 102251, 31 pp. [DOI:10.1016/j.aam.2021.102251]
- C. Bhattacharjee and I. Molchanov, Gaussian approximation for sums of region-stabilizing scores. Electr. J. Probab., 27, Paper No. 111: 1-27, 2022.
- Z. Kabluchko, A. Marynych and I. Molchanov. Generalised convexity with respect to families of affine maps. Israel J. Math. 2023. To appear

## Group members

- Name / Titel
- Prof. Dr. Ilya Molchanov
- Funktion
- Professor of Probability Theory / Co-director of IMSV
- ilya.molchanov@unibe.ch

- Phone
- +41 31 684 88 01

- Name / Titel
- Tommaso Visonà
- Funktion
- Assistant III / PhD student
- tommaso.visona@unibe.ch

- Phone
- +41 31 684 54 62

#### Former members

- Dr. Riccardo Turin
- Dr. Qiyu Li
- Dr. Michael Schmutz
- Dr. Federico Pianoforte
- Dr. Michael Mayer
- Dr. Chris Kopp