A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence,

We consider a stationary time series {Xt} given byXt=[summation operator][infinity]k=-[infinity] [psi]kZt-k, where {Zt} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf([lambda]) of {Xt} is squared integrable andm[tau] [summation operator]k[greater-or-equal, slanted]m [psi]2k-->0 for some[tau]>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionm[tau] [summation operator]k[greater-or-equal, slanted]m [psi]2k-->0 for some[tau]>1/2 is equivalent to the squared integrability off([lambda]). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm [summation operator]k[greater-or-equal, slanted]m [psi]2k-->0.