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Convex hulls of random walks: expected number of faces and face probabilities

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posted on 2023-06-09, 08:55 authored by Zakhar Kabluchko, Vladislav VysotskiyVladislav Vysotskiy, Dmitry Zaporozhets
Consider a sequence of partial sums Si=?1+…+?i, 1=i=n, starting at S0=0, whose increments ?1,…,?n are random vectors in Rd, d=n. We are interested in the properties of the convex hull Cn:=Conv(S0,S1,…,Sn). Assuming that the tuple (?1,…,?n) is exchangeable and a certain general position condition holds, we prove that the expected number of k-dimensional faces of Cn is given by the formula E[fk(Cn)]=2·k!n!?l=08[n+1d-2l]{d-2lk+1}, for all 0=k=d-1, where [nm] and {nm} are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0=i1<…

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Advances in Mathematics

ISSN

0001-8708

Publisher

Elsevier

Volume

320

Page range

595-629

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-11-20

First Open Access (FOA) Date

2018-09-13

First Compliant Deposit (FCD) Date

2017-11-20