Asymptotic Properties of Serial Covariances for Nonlinear Stationary Processes

Let {Xt; t ∈ Z} be a strictly stationary process with mean zero and autovariance function (a.c.v.f.) γv, v ∈ Z. Let γ̂v = n − 1 ∑n − |v|t = 1 be the serial covariance of order v computed from a sample X1, ..., Xn drawn from {Xt}. We assume that {Xt} is nonlinear but satisfies some mild regularity conditions. We prove that for a fixed integer l, the distribution of n1/2(γ̂v − γv), ..., n1/2(γ̂v+l − γv + l) is, asymptotically, normal with mean zero and a finite covariance matrix. The result holds both for finite v and when v → ∞ but v/n → 0 as n → ∞.