Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks
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Publication:6368395
DOI10.1007/S00440-022-01146-9zbMath1526.60012arXiv2105.11365WikidataQ114229356 ScholiaQ114229356MaRDI QIDQ6368395
Alexander Marynych, Zakhar Kabluchko
Publication date: 24 May 2021
Abstract: Let $xi_1,xi_2,ldots$ be a sequence of independent copies of a random vector in $mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=xi_1+cdots+xi_i$, and let $C_{n,d}:= ext{conv}(0,S_1,S_2,ldots,S_n)$ be the convex hull of the first $n+1$ points it has visited. The polytope $C_{n,d}$ is called $k$-neighborly if for every indices $0leq i_0 <cdots < i_kleq n$ the convex hull of the $k+1$ points $S_{i_0},ldots, S_{i_k}$ is a $k$-dimensional face of $C_{n,d}$. We study the probability that $C_{n,d}$ is $k$-neighborly in various high-dimensional asymptotic regimes, i.e. when $n$, $d$, and possibly also $k$ diverge to $infty$. There is an explicit formula for the expected number of $k$-dimensional faces of $C_{n,d}$ which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of $C_{n,d}$. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.
Geometric probability and stochastic geometry (60D05) Sums of independent random variables; random walks (60G50) Partitions of sets (05A18) Bell and Stirling numbers (11B73) Combinatorial probability (60C05) Asymptotic enumeration (05A16) Random convex sets and integral geometry (aspects of convex geometry) (52A22)
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