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

We consider a stationary time series {Xt} given by Xt = ΣkψkZt − k, where the driving stream {Zt} consists of independent and identically distributed random variables with mean zero and finite variance. Under the assumption that the filtering weights ψk are squared summable and that the spectral density of {Xt} 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 {ψk] is summable.