An invariance principle for associated random fields (Q1106543)

An invariance principle is given for stationary, associated random measures X on R d satisfying for some \(\delta >0\) and K, \[ E(| X(B)- E X(B)|^{2+\delta})\leq K| B|^{1+\delta /2} \] for all rectangular boxes \(B\supset [0,1]\) d (\(| B|\) is the Lebesgue measure of B). It is proved that this applies to Poisson center cluster random measures X with centers U and members \(V=\{V_ x:x\in R\) \(d\}\) such that \(E| V_ x(R\) \(d)|^{2+\delta}<\infty.\) When \(d=1\) or 2 the invariance principle holds assuming only that X has summable covariances. For this, see the first author and \textit{E. Waymire}'s article in Ann. Probab. 13, 1267-1278 (1985; Zbl 0579.60039); it follows by applying results of \textit{C. M. Newman} and \textit{A. L. Wright} [ibid. 9, 671-675 (1981; Zbl 0465.60009) and Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 361-371 (1982; Zbl 0465.60010)].