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

摘要

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