LIMITING RESULTS FOR ARRAYS OF BINARY RANDOM VARIABLES ON RECTANGULAR LATTICES UNDER SPARSENESS CONDITIONS.

Limiting results are given for arrays right brace X//i//j(m,n): (i,j) are elements of D//m//n left brace of binary random variables distributed as particular types of Markov random fields over m multiplied by n rectangular lattices D//m//n. Under some sparseness conditions which restrict the number of X//i//j(m,n)'s which are equal to one it is shown that the random variables (l equals 1, multiplied by (times) multiplied by (times) multiplied by (times) ,r) converge to independent Poisson random variables for 0 less than d//1 less than d//2 less than multiplied by (times) multiplied by (times) multiplied by (times) less than d, when m and n approach infinity . 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.