On processes with hyperbolically decaying autocorrelations

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d ∈ (−∞, 0), which enjoys non-summable partial autocorrelations and autocorrelations decaying as rapidly as ρn ≅ n−1+2d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).