We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over a binomial point process of n points in a toroidal spatial region of volume n. The proof is based on Stein's method via size-biased couplings. 1. Introduction. Given a collection of n independent uniform...

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H0 : f = f0, where the alternative H1 is expressed with respect to L2-norm (i.e. has the form ψ−2 n ‖f −f0‖2 ≥ C). Our procedure is adaptive with res...

A classical theorem of Rényi [26] for the number of empty boxes, denoted by μ0(n,M), in a sequence of n random allocations of indistinguishable balls into M boxes with equal probability 1/M , can be stated as follows: If the variance of μ0(n,M) tends to infinity with n then μ0(n,M) is asymptotically normally distributed. This result, seldom stated in this form in the literature, was proved by R...

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry-Esseen bounds and local limit theorems for the norm...

In this paper, we revisit two multi-terminal lossy source coding problems: the lossy source coding problem with side information available at the encoder and one of the two decoders, which we term as the Kaspi problem (Kaspi, 1994), and the multiple description coding problem with one semi-deterministic distortion measure, which we refer to as the Fu-Yeung problem (Fu and Yeung, 2002). For the ...