No abstract

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on R d are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.

The behavior of the Kozachenko -Leonenko estimates for the (differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L 2 -consistency of the estimates are established. The conditions employed involve the analogues of the Hardy -Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.