Strong limit theorems for quasi-orthogonal random fields
A double sequence \(\{X_{ik};i,k\geq 1\}\) of random variables is called quasi-orthogonal if a real double sequence \(\rho\) (m,n) exists such that \[ | E X_{ik}X_{j\ell}| \leq \rho (| i-j|,| k-\ell |)\sigma_{ik}\sigma_{j\ell}\quad for\quad all\quad i,j,k,\ell \geq 1, \] where \(\sigma^ 2_{ik}\) is the variance of \(X_{ik}\). For non- decreasing sequences \(\{\lambda_ 1(n)\}\) and \(\{\lambda_ 2(n)\}\), let \(\xi_{mn}=\sum^{m}_{i=1}\sum^{n}_{k=1}X_{ik}/(\lambda_ 1(m)\lambda_ 2(n)).\) Conditions are given which ensure that \(\xi_{mn}\to 0\) almost surely (a.s.) as \(m+n\to \infty.\) Similar results are proved for subsequences of \(\{\xi_{mn}\}\). In addition, sufficient conditions for \[ \lim_{n\to \infty}a_ n^{- 1}\sum_{(i,k)\in Q_ n}X_{ij}=0\quad a.s. \] are presented, where \(Q_ 1\subset Q_ 2\subset..\). are non-empty sets of lattice points in the first quadrant. The rates of these convergences are also explored.
scientific article; zbMATH DE number 4122960
scientific article