Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Let ξ 1 , ξ 2 , … be a sequence of independent copies of a random vector in R d having an absolutely continuous distribution. Consider a random walk S i : = ξ 1 + ⋯ + ξ i , and let C n , d : = conv ( 0 , S 1 , S 2 , … , S n ) be the convex hull of the first n + 1 points it has visited. The polytope...

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Bibliographic Details
Published in:	Probability theory and related fields Vol. 184; no. 3-4; pp. 969 - 1028
Main Authors:	Kabluchko, Zakhar, Marynych, Alexander
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022 Springer Nature B.V
Subjects:	Central limit theorem Combinatorial analysis Composition Computational geometry Convexity Economics Finance Hulls Insurance Management Mathematical and Computational Biology Mathematical and Computational Physics Mathematics Mathematics and Statistics Operations Research/Decision Theory Phase transitions Polytopes Probability Probability Theory and Stochastic Processes Quantitative Finance Random walk Statistics for Business Theoretical Random compositions 60C05 Weyl chambers 26C10 52A22 52A23 Random polytopes Stirling numbers 30C15 Random walks function Primary: 11B73 Records 60F05 Convex hulls 60F10 Lah distribution Large deviations Conic intrinsic volumes Lambert vectors Lah numbers Mod-Poisson convergence Central limit theorem Threshold phenomena Neighborliness 05A16 Secondary: 60D05 05A18
ISSN:	0178-8051, 1432-2064
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Summary:	Let ξ 1 , ξ 2 , … be a sequence of independent copies of a random vector in R d having an absolutely continuous distribution. Consider a random walk S i : = ξ 1 + ⋯ + ξ i , and let C n , d : = conv ( 0 , S 1 , S 2 , … , 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 any indices 0 ≤ i 1 < ⋯ < i k ≤ n the convex hull of the k points S i 1 , … , S i k is a ( k - 1 ) -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 ∞ . There is an explicit formula for the expected number of ( k - 1 ) -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.
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0178-8051 1432-2064
DOI:	10.1007/s00440-022-01146-9