On permissible correlations for locally correlated stationary processes (Q1344830)

Every second-order stationary process with index set \(\{0, \pm1, \pm2, \dots\}\) and zero autocorrelations at lags greater than one can be represented as MA(1). The situation is more complicated if one considers an analogy for a stationary process which is indexed by the two- dimensional integer lattice. The paper establishes necessary and sufficient conditions on real numbers \(\{\rho_ 1, \rho_ 2, \rho_ 3, \rho_ 4\}\) to ensure the existence of a locally correlated process \(\{ X(t_ 1, t_ 2)\}\) with correlations \(\{\rho_ 1, \rho_ 2, \rho_ 3, \rho_ 4\}\) (this process fulfills \(\rho(1,0)= \rho(-1,0)= \rho_ 1\), \(\rho(1,1)= \rho(-1, -1)= \rho_ 2\), \(\rho(0,1)= \rho(0, -1)= \rho_ 3\), \(\rho(1, -1)= \rho(-1, 1)= \rho_ 4\), \(\rho(i, j)=0\) otherwise).