# 21 квітня 2021 року о 14 год 15 хв

Let $$X_1,X_2,\ldots$$ be independent random vectors in $$\mathbb{R}^d$$ having an absolutely continuous distribution. Consider the random walk $$S_k:=X_1+\ldots+X_k$$, and let $$P_n:={\rm conv}\{0,S_1,S_2,\ldots,S_n\}$$ be the convex hull of its first $$n$$ steps. We shall be interested in the number of the $$k$$-dimensional faces of the polytope $$P_n$$ and in particular, whether this number is equal to the maximal possible number $$\binom {n+1}{k+1}$$ with high probability, as $$n$$, $$d$$, and possibly also $$k$$ diverge to $$\infty$$. There is an explicit formula for the expected number of $$k$$-dimensional faces which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties.

Доповідач: Zakhar Kabluchko (University of Münster, Germany)

Дата проведення: 21 квітня 2021 року о 14 год 15 хв.

### Новини кафедри

Joan Miró (Жуан Міро)

Personnage devant le soleil (Характер перед сонцем), 1968

Помічено Д. Загребельною в одному з музеїв Барселони