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

Abstract

We consider a stationary time series {Xt} given byXt=∑∞k=−∞ψkZt−k, where {Zt} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {Xt} is squared integrable andmτ∑|k|⩾mψ2k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionmτ∑|k|⩾mψ2k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm∑|k|⩾mψ2k→0.