Lah distributions, Stirling numbers, and phase transitions for convex hulls of random walks


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.