Normal approximations for binary lattice systems

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.