Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Abstract

Schreiber and Yukich [Ann. Probab. 36 (2008) establish an asymptotic representation for random convex polytope geometry in the unit ball B d , d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

Normal approximation for stabilizing functionals

“…. ,ξ d−1 can be shown similarly as in Lemma 7.1 of[9]. Similar considerations as in the proof of Lemma 5.10 show thatξ d,s (x, P s ∪ {x} ∪ A) ≤ sQ(A x,R(x,Ps∪{x}) ).Combining this with Lemma 5.7 and Lemma 5.11 leads to the inequality forξ d,s in the Poisson case, which can be proven similarly in the binomial case.…”

“…. , V d } and A is the unit ball, Theorem 7.1 of [9] gives a central limit theorem for h(Conv(P s )), with convergence rates involving extra logarithmic factors. We are unaware of central limit theorem results for intrinsic volume functionals over binomial input.…”

“…, d − 1}, in the form of (1.1) and (1.2), we need some more notation. Given the convex set A and a linear subspace E, denote by A|E the orthogonal projection of A onto E. and, as in [9], define the projection avoidance function θ A,M j :…”

Gaussian polytopes: A cumulant-based approach

“…In particular, for any fixed λ there is no centred ball with radius only depending on λ (or any other deterministic set that depends on the parameter λ only) in which a Gaussian polytope is included with probability one. This in turn implies that the scaling transformation we borrow from [10], which we recall in Section 2 below, maps a Gaussian polytope into a random set in the product space R d−1 × R, while the scaling transformation for random polytopes in the unit ball has R d−1 × [0, ∞) as its target space, see [9]. Here, the upper half-space R d−1 × [0, ∞) corresponds to the image of an appropriate centred ball that contains the Gaussian polytope with high probability, while the lower half-space R d−1 × (−∞, 0) corresponds to the image of its complement.…”

Multivariate Normal Approximation for Functionals of Random Polytopes

“…Functionals like the intrinsic volumes V j (K t ) and the components f j (K t ) of the f -vector, see Section 3, of the random polytope K t have been studied prominently, see [CY14; Rei10; CSY13, Section 1] and the references therein as well as the remarks and references on [LSY17,Theorem 5.5] for more details. Central limit theorems for V j (K t ) were proven in the special case that K is the ddimensional Euclidean unit ball, see [CSY13] and [Sch02]. Short proofs for the binomial case K n , where n i.i.d.…”