Asymptotic Properties of Serial Covariances for Nonlinear Stationary Processes

Let {Xt; t [set membership, variant] 0} be a strictly stationary process with mean zero and autovariance function (a.c.v.f.) [gamma]v, v [set membership, variant] 0. Let [gamma]v = n - 1 [summation operator]n - vt = 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([gamma]v - [gamma]v), ..., n1/2([gamma]v+l - [gamma]v + l) is, asymptotically, normal with mean zero and a finite covariance matrix. The result holds both for finite v and when v --> [infinity] but v/n --> 0 as n --> [infinity].