The Asymptotic Distribution of Sample Autocorrelations for a Class of Linear Filters

We consider a stationary time series {X_t} given by X_t = Σ_kψ_kZ_{t − k}, where the driving stream {Z_t} consists of independent and identically distributed random variables with mean zero and finite variance. Under the assumption that the filtering weights ψ_k are squared summable and that the spectral density of {X_t} is squared integrable, it is shown that the asymptotic distribution of the sequence of sample autocorrelation functions is normal with covariance matrix determined by the well-known Bartlett formula. This result extends classical theorems by Bartlett (1964, J. Roy Statist. Soc. Supp.8 27-41, 85-97) and Anderson and Walker (1964, Ann. Math. Statist.35 1296-1303), which were derived under the assumption that the filtering sequence {ψ_k] is summable.