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

✍ Scribed by R. Cavazoscadena

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 639 KB
• Volume: 48
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1994.1005

✦ Synopsis

We consider a stationary time series (\left{X_{t}\right}) given by (X_{1}=\sum_{k} \psi_{k} Z_{l-k}), where the driving stream (\left{Z_{i}\right}) consists of independent and identically distributed random variables with mean zero and finite variance. Under the assumption that the filtering weights (\psi_{k}) are squared summable and that the spectral density of (\left{X_{i}\right}) 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 (\left{\psi_{k}\right}) is summable. 1994 Academic Press, Inc.