First-order planar autoregressive model

This paper establishes the conditions of existence of a stationary solution to the first order autoregressive equation on a plane as well as properties of the stationarity solution. The first-order autoregressive model on a plane is defined by the equation X-i,X-j=aX(i-1),(j)+bX(i,j-1)+cX(i-1),(j-1)+& varepsilon;(i,j). A stationary solution X to the equation exists if and only if (1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)>0. The stationary solution X satisfies the causality condition with respect to the white noise & varepsilon; if and only if 1-a-b-c>0, 1-a+b+c>0, 1+a-b+c>0 and 1+a+b-c>0. A sufficient condition for X to be purely nondeterministic is provided. An explicit expression for the autocovariance function of X at some points is provided. With Yule-Walker equations, this allows to compute the autocovariance function everywhere. In addition, all situations are described where different parameters determine the same autocovariance function of X.