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Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions

Published online by Cambridge University Press:  14 July 2016

Roy Saunders ,
Richard J. Kryscio  and
Gerald M. Funk
Roy Saunders*
Affiliation:
Northern Illinois University
Richard J. Kryscio*
Affiliation:
Northern Illinois University
Gerald M. Funk*
Affiliation:
Northern Illinois University
*
∗ Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
∗ Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
∗ Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
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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 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.

Keywords

MARKOV RANDOM FIELDLIMITING DISTRIBUTIONBINARY VARIABLESSPARSENESSCLUSTERINGLATTICEPOISSONNEAREST NEIGHBOR
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 554 - 566
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research partially supported by the National Science Foundation Grant No. MCS 77–03582.

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