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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