81,916 research outputs found

    Probability that n random points are in convex position

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    We show that n random points chosen independently and uniformly from a parallelogram are in convex position with probability

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

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    Consider a sequence of partial sums Si=Ī¾1+ā€¦+Ī¾i, 1ā‰¤iā‰¤n, starting at S0=0, whose increments Ī¾1,ā€¦,Ī¾n are random vectors in Rd, dā‰¤n. We are interested in the properties of the convex hull Cn:=Conv(S0,S1,ā€¦,Sn). Assuming that the tuple (Ī¾1,ā€¦,Ī¾n) 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)]=2ā‹…k!n!āˆ‘l=0āˆž[n+1dāˆ’2l]{dāˆ’2lk+1}, for all 0ā‰¤kā‰¤dāˆ’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 0ā‰¤i1<ā€¦<ik+1ā‰¤n, the points Si1,ā€¦,Sik+1 form a k-dimensional face of Conv(S0,S1,ā€¦,Sn). This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments Ī¾k's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types Anāˆ’1 and Bn. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position

    Band depths based on multiple time instances

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    Bands of vector-valued functions f:Tā†¦Rdf:T\mapsto\mathbb{R}^d are defined by considering convex hulls generated by their values concatenated at mm different values of the argument. The obtained mm-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for m=1m=1) to the convex hull in the functional space if the number mm of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions.Comment: 12 page
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