Central limit theorem for Baxter sums of Gaussian random fields (Q1600591)

For a Gaussian random field \(X(t)\), \(t \in[0,1]^d\), with zero mean, sums of squares of increments (Baxter sums) are considered of the form \(S_n=a_n^{-d}\sum(\Delta^p X_k/\sigma_k)^2\), where \(p=(p_1,\dots,p_d)\), \(k=(k_1,\dots,k_d)\), \(\Delta^p X_k\) is an increment of \(X\) of order \(p\) at the point \(a_n^{-1}k\) with step \(a_n\), \(\sigma_k^2={\mathbf E}(\Delta^p X_n)^2\). Conditions of asymptotic normality of \((S_n-1)/\sqrt{\text{Var} S_N}\) are derived. As a corollary the author obtains conditions of a.s. convergence of \(S_n \to 1\) (a generalization of Baxter theorem). These results are applied to the estimation of some parameter of the correlation function of \(X\).