QUENOUILLE-TYPE THEOREM ON AUTOCORRELATIONS

SIMON KU AND EUGENE SENETA

School of Mathematics and Statistics, The University of Sydney, N.S.W. 2006, Australia

(Received May 22, 1995; revised February 1, 1996)

Abstract. The central result is a limit theorem for not necessarily stationary processes resembling AR(p). Assumption of a vector limit distribution for standardized sample autocorrelations leads to the convergence of a vector limit distribution for ordinary sample partial autocorrelations, and to a clear relationship between the two limit distributions. The motivation is the study of the case p = 1 by Mills and Seneta (1989, Stochastic Process Appl., 33, 151–161). The central result is used to explain the nature of the relationship between the two results of Quenouille in the classical stationary AR(p) setting.

Key words and phrases: Limit theorem, sample autocorrelations, sample partial autocorrelations, Quenouille’s test.