Two Kinds Of Limit Results For Random Fields

The paper studies two kinds of limit results for random fields withmulti-dimensional index. Firstly, the paper discusses the laws of the logarithm for theweighted sums of the random fields of independent and identically distributed randomvariables with multi-dimensional index, a sharp upper bound on the laws of thelogarithm is obtained, which improves the well-known result. As the application, theconvergence rate for the nonparametric regression estimators is obtained. Secondly,under the conditions of Kolmogorov-Chung’s strong law of large numbers, theconvergence rate of the strong law of large numbers for Cesàro summation of therandom fields of pairwise independent random variables with two-dimensional indexis obtained, as the application, the convergence rate of the strong law of largenumbers for Cesàro summation of the random fields of pairwise independent andidentically distributed random variables with two-dimensional index is obtained,which is as same as the independent case.