Central Limit Theorems for Associated Random Variables and the Percolation Model

We prove a central limit theorem for families ₙ(N): n ∈ Λ(N)\ of associated random variables indexed by subsets Λ(N) of Zᵈ, as N → ∞; this is an extension of the Newman-Wright invariance principle for associated stationary sequences ₙ: n ≥ 1\ satisfying ∑ₙ cov(X₁, Xₙ) < ∞, but with the stationarity property replaced by conditions on the moments of the X's. The theorem has applications to the voter model and the percolation model. In the latter case, it provides an extension of a central limit theorem of the authors [4], by reducing the severity of the moment conditions. Also, we prove a central limit theorem for certain non-stationary non-associated families of random variables which arise in percolation theory. This includes, for example, a central limit theorem for the number of open clusters contained within the circuit γ(n) of Z², where \γ(n)\ is a sequence of circuits which satisfy a regularity condition and whose interiors \∘γ(n)\ satisfy |∘γ(n)| → ∞ as n → ∞.