Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

Schreiber, Tomasz; Yukich, J. E.

doi:10.1214/009117907000000259

Cited by 42 publications

(91 citation statements)

References 36 publications

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“…, d − 1), can be estimated using the above methods as shown in [7] and [13]. Very recently, Schreiber and Yukich [18] determined the variance of f 0 (K n ) asymptotically when K is the unit ball, a significant breakthrough. Hopefully, their methods can work for all f s (K n ) and V s (K n ) as well.…”

Section: History and Resultsmentioning

confidence: 99%

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010

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Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by K n the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes V s (K n ) of K n for s ∈ {1, . . . , d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of K n . The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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Section: History and Resultsmentioning

confidence: 99%

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010

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“…General intrinsic volumes are treated in [11] under a weak smoothness assumption. Recently, even variance estimates, laws of large numbers and central limit theorems have been proved in different settings in a sequence of contributions, for instance by Bárány, Reitzner, Schreiber, Vu and Yukich; see [3,5,29,30,39,44,45]. For more details on the current state-of-the-art of this line of research, see the survey papers by Weil and Wieacker [46], Gruber [16] and Schneider [36], and the recent monograph of Schneider and Weil [37].…”

Section: Introductionmentioning

confidence: 99%

The mean width of random polytopes circumscribed around a convex body

Böröczky

Fodor²,

Hug

2010

Journal of the London Mathematical Society

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Let K be a d-dimensional convex body and let K (n) be the intersection of n halfspaces containing K whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of K (n) and K, and another asymptotic formula for the expectation of the number of facets of K (n) . These results are achieved by establishing an asymptotic result on weighted volume approximation of K and by 'dualizing' it using polarity.

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“…In view of the exact formulae for E N n+1 , E N n+2 , and E N 2 n+2 arising from Theorem 1, we obtain, as immediate consequences of Theorem 2, the exact formulae This shows that A(T ) −2 var D n ∼ 28 27 n −2 log n as n → ∞. Proof of Theorem 2.…”

Section: Theorem 2 the Expected Value And The Variance Of D N Are Gimentioning

confidence: 60%

Exact Formulae for Variances of Functionals of Convex Hulls

Buchta

2013

Adv. Appl. Probab.

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The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider-in view of affine invariance-n points P 1 , . . . , P n distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number N n of points among P 1 , . . . , P n , which are vertices of the convex hull of (0, 1), P 1 , . . . , P n , and (1, 0). Correspondingly, D n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var N n and var D n . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev andKhamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

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Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

Cited by 42 publications

References 36 publications

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

The mean width of random polytopes circumscribed around a convex body

Exact Formulae for Variances of Functionals of Convex Hulls

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