A NOTE ON THE EMBEDDING OF DISCRETE‐TIME ARMA PROCESSES

Abstract

Abstract. Let {Xn, n= 0, 1, 2,…} be a discrete‐time ARMA(p, q) process with q < p whose autoregressive polynomial has r (not necessarily distinct) negative real roots. According to a recent result of He and Wang (On embedding a discrete‐parameter ARMA model in a continuous‐parameter ARMA model. J. Time Ser. Anal. 10 (1989), 315–23) there exists a continuous‐time ARMA (p', q') process {Y(t), t≥0} with q' < p'=p+r such that {Y(n), n= 0, 1, 2,…} has the same autocorrelation function as {Xn}. In this paper we show that this result is false by considering the case when {Xn} is a discrete‐time AR(2) process whose autoregressive polynomial has distinct complex conjugate roots. We identify the proper subset of such processes which are embeddable in a continuous‐time ARMA(2, 1) process. We show that every discrete‐time AR(2) process with distinct complex conjugate roots can be embedded in either a continuous‐tie ARMA(2, 1) process or a continuous‐time ARMA(4, 2) process, or in some cases both. We derive an expression for the spectral density of the process obtained by sampling a general continuous‐time ARMA(p, q) process (with distinct autoregressive roots) at arbitrary equally spaced time points. The expression clearly shows that the sampled process is a discrete‐time ARMA (p', q') process with q' < p.