A ug 2 01 1 Local central limit theorems in stochastic geometry Mathew

F eb 2 01 5 CENTRAL LIMIT THEOREMS FOR SOME SET PARTITION STATISTICS

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1, 2, . . . , n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

Limit theorems in discrete stochastic geometry

This overview surveys two general methods for establishing limit theorems for functionals in discrete stochastic geometry. The functionals of interest are linear statistics with the general representation ∑x∈X ξ (x,X ), where X is locally finite and where the interactions of x with respect to X , given by ξ (x,X ), exhibit spatial dependence. We focus on subadditive methods and stabilization me...

Central Limit Theorems for Some Graphs in Computational Geometry

Let Bn be an increasing sequence of regions in d-dimensional space with volume n and with union d. We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to Bn, or by by taking n uniformly distributed points in Bn. The sets Bn could be all cubes but a more general class of regions Bn is considered. Using this general ...

Asymptotic Expansions for Stochastic Processes

The central limit theorems are the basis of the large sample statistics. In estimation theory, the asymptotic efficiency is evaluated by the asymptotic variance of estimators, and in testing statistical hypotheses, the critical region of a test is determined by the normal approximation. Though asymptotic properties of statistics are based on central limit theorems, the accuracy of their approxi...

Central and Local Limit Theories in Markov Dependent Random Variables