Consider an array of binary random variables distributed over an m 1(n) by m 2(n) rectangular lattice and let Y 1(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) = (Y 1(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.

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.

In this article we present a limiting result for the random variable Yn(r) which arises in a clustering model of Strauss (1975). The result is that under some sparseness-of-points conditions the process {Yn(r): 0 ≦ r ≦ r∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r∞} when n → ∞. Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed.