Statistical limit theorems for weak dependent random fields

Real and vector-valued weakly dependent random fields are investigated. The main sources of interest here are percolation theory, statistical physics, mathematical statistics and reliability theory. For such random fields the consistency of statistics with local averaging that were introduced by Peligrad, Shao, Bulinski and Vronski is proved. Statistical version of the central limit theorem (CLT) with random matrix normalization (involving these statistics) is established. The main result of the first Chapter provides an estimate of the convergence rate in this CLT over a family of convex bounded sets belonging to $\R^k$.
Multidimensional analogues of Parzen’s and Rosenblatt’s kernel estimators of the long-run covariance matrix are constructed for centered weakly dependent and not necessarily stationary random fields. For sequences of random vectors, possessing mixing property, the estimators of this type were studied by White, Hansen and Andrews. Among the results of this Chapter we mention consistency and strong consistency of estimators under consideration.
Moreover in the third Chapter new moment and maximal inequalities are obtained for sums of dependent multiindexed random variables. The proofs of these theorems are essentially based on the Moricz method and the recent author’s results on separation of discrete sets in a multidimensional space that develop the technique introduced by Bernstein and Lifshits.