Results and methods discussed in two previous papers are extended to other cases. Comparison is made with recent work by Pickard (1980) and an earlier conjecture is disproved.

Consider an array of binary random variables distributed over an m1(n) by m2(n) rectangular lattice and let Y1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y1(n), · ··, Yr(n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

Sufficient conditions are derived for ergodicity of the two-dimensional process of binary variables discussed by Galbraith and Walley (1976).

In this article we give limiting results for arrays {Xij (m, n) (i, j) Dmn} of binary random variables distributed as particular types of Markov random fields over m x n rectangular lattices Dmn. Under some sparseness conditions which restrict the number of Xij (m, n)'s which are equal to one we show that the random variables (l = 1, ···, r) converge to independent Poisson random variables for 0 < d1 < d2 < · ·· < dr when m→∞ nd∞. The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.

We further consider a two-dimensional stochastic process of binary variables previously discussed by Galbraith and Walley (1976). A method of estimating properties of the equilibrium distribution is described which considerably extends the region of probability space that can satisfactorily be explored.

We consider a two-dimensional stochastic process of binary variables xij defined so that the conditional distribution of xij given its predecessors depends only on xi–1j and xij–1 Properties of the equilibrium distribution, when this exists, are investigated using three different representations of the process and explicit results are given in some special cases.