Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions
- 1 September 1979
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 16 (3) , 554-566
- https://doi.org/10.2307/3213084
Abstract
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 overmxnrectangular latticesDmn.Under some sparseness conditions which restrict the number ofXij(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< · ·· <drwhenm→∞ 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.Keywords
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