Normal approximation for stabilizing functionals

Abstract

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind.Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of R d , including m-dimensional Riemannian manifolds, m ≤ d. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the k-face and ith intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension d ≥ 2. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

Multivariate second order Poincaré inequalities for Poisson functionals

“…This is the advantage of these findings over Malliavin-Stein bounds for normal approximation of Poisson functionals which either require the knowledge of the chaos expansion of F (see, for example, [7,12,23,30]) or which involve bounds expressed in terms of gradient operators and conditional expectations as in [25]. Inequality (1.5) yields rates of normal approximation for some classic problems in stochastic geometry and some non-linear functionals of Poisson-shot-noise processes [16], as well as for functionals of convex hulls of random samples in a smooth convex body, statistics of nearest neighbors graphs, the number of maximal points in a random sample, and estimators of surface area and volume arising in set approximation [15]. The rates of convergence for these examples are presumably optimal.…”

“…The set-up of Corollary 1.4, in which one re-scales by the square root of the intensity parameter and in which the (6 + ε)-th moments of the un-rescaled difference operators are bounded, frequently occurs in problems in stochastic geometry; see e.g. [15,16].…”

Normal convergence of nonlocalised geometric functionals and shot-noise excursions

“…4. Similar results where the input consists of m n iid variables uniformly distributed inW n , with m n = |W n |, should be within reach by applying the results of [15], following a route similar to [16].…”

Multivariate Normal Approximation for Functionals of Random Polytopes

“…The investigation of random convex hulls is one of the classical problems in stochastic geometry, see for instance the survey article [Hug13] and the introduction to stochastic geometry [HR16]. 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].…”

“…uniformly distributed points in K are considered instead of a Poisson point process, were derived in [TTW18]. Recently [LSY17] embedded the problem for both cases, the binomial and the Poisson case, in the theory of stabilizing functionals deriving central limit theorems for smooth convex bodies removing the logarithmic factors in the error of approximation improving the rate of convergence. We extend the results of [TTW18] on the intrinsic volumes to the more general case of continuous and motion invariant valuations.…”