Limit Theorems in Discrete Stochastic Geometry
We survey two general methods for establishing limit theorems for functionals in discrete stochastic geometry. The functionals are linear statistics with the general representation \(\sum _{x\in \mathcal{X}}\xi (x,\mathcal{X})\) , where \(\mathcal{X}\) is
- PDF / 429,350 Bytes
- 37 Pages / 439.36 x 666.15 pts Page_size
- 88 Downloads / 221 Views
Limit Theorems in Discrete Stochastic Geometry Joseph Yukich
Abstract We survey two general methods for establishing limit theorems for functionals in discrete stochasticPgeometry. The functionals are linear statistics with the general representation x2X .x; X /, where X is finite and where the interactions of x with respect to X , given by .x; X /, are spatially correlated. We focus on subadditive methods and stabilization methods as a way to obtain weak laws of large numbers, varianceP asymptotics, and central limit theorems for normalized and re-scaled versions of niD1 .i ; fj gnj D1 /, where j , j 1, are i.i.d. random variables. The general theory is applied to deduce the limit theory for functionals arising in Euclidean combinatorial optimization, convex hulls of i.i.d. samples, random sequential packing, and dimension estimation.
8.1 Introduction This overview surveys two general methods for establishing limit theorems, including weak laws of large numbers, variance asymptotics, and central limit theorems, for functionals of large random geometric structures. By geometric structures, we mean for example networks arising in computational geometry, graphs arising in Euclidean optimization problems, models for random sequential packing, germgrain models, and the convex hull of high density point sets. Such diverse structures share only the common feature that they are defined in terms of random points belonging to Euclidean space Rd . The points are often the realization of i.i.d. random variables, but they could also be the realization of Poisson point processes or even Gibbs point processes. There is scope here for generalization to functionals of point processes in more general spaces, including manifolds and general metric
J. Yukich () Lehigh University, Bethlehem, PA 18015, USA e-mail: [email protected] E. Spodarev (ed.), Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics 2068, DOI 10.1007/978-3-642-33305-7 8, © Springer-Verlag Berlin Heidelberg 2013
239
240
J. Yukich
spaces, but for ease of exposition we shall usually restrict attention to point processes in Rd . As such, this introductory overview makes few demands involving prior familiarity with the literature. Our goals are to provide an accessible survey of asymptotic methods involving (a) subadditivity and (b) stabilization and to illustrate the applicability of these methods to problems in discrete stochastic geometry. The treatment of subadditivity parallels that in [524].
8.1.1 Functionals of Interest Functionals of geometric structures are often formulated as linear statistics on finite point sets X of Rd , that is to say consist of sums represented as H.X / WD H .X / WD
X
.x; X /;
(8.1)
x2X
where the function , defined on all pairs .x; X /, x 2 X , represents the interaction of x with respect to input X . The focus of this chapter is to develop the large n limit theory for the normalized sums n1 H .fi gniD1 /;
(8.2)
where i ; i 1; are i.i.d. with values in Œ0; 1d .
Data Loading...
