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Normal approximations for binary lattice systems

Published online by Cambridge University Press:  14 July 2016

Richard J. Kryscio ,
Roy Saunders  and
Gerald M. Funk
Richard J. Kryscio*
Affiliation:
Northern Illinois University
Roy Saunders*
Affiliation:
Northern Illinois University
Gerald M. Funk*
Affiliation:
Loyola University
*
∗Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, U.S.A.
∗Postal address: Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, U.S.A.
∗∗Postal address: Department of Mathematical Sciences, Loyola University of Chicago, 6525 North Sheridan Rd., Chicago, IL 60626, U.S.A.
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Abstract

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.

Keywords

MARKOV RANDOM FIELDDISCRETELATTICEBINARY VARIABLESNEAREST NEIGHBORCLUSTERINGUNIFORM PACKINGNORMAL DISTRIBUTIONLIMITING DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 674 - 685
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research supported in part by NSF Grant No. MCS 77–03582.

References

Besag, J. E. (1974) Spatial interaction and the statistical analysis of lattice systems. J. R. Statist. Soc. B 36, 192–235.Google Scholar
Bloemena, A. R. (1964) Sampling From a Graph. Mathematical Centre Tracts 2, Amsterdam.Google Scholar
David, F. N. (1970) Measurement of diversity, I. 6th Berkeley Symp. Math. Statist. Prob. , 631–648.Google Scholar
Feller, W. (1967) An Introduction to Probability Theory and its Applications , Volume I, 3rd ed. Wiley, New York.Google Scholar
Fraser, D. A. S. (1957) Nonparametric Methods in Statistics. Wiley, New York.Google Scholar
Lehmann, E. L. (1959) Testing Statistical Hypotheses. Wiley, New York.Google Scholar
Saunders, R., Kryscio, R. J. and Funk, G. M. (1979) Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions. J. Appl. Prob. 16, 554–566.Google Scholar
Strauss, D. J. (1975) Analyzing binary lattice data with the nearest neighbor property. J. Appl. Prob. 12, 702–712.Google Scholar
Strauss, D. J. (1977) Clustering on coloured lattices. J. Appl. Prob. 14, 135–143.CrossRefGoogle Scholar