What does the partial autocorrelation function look like for large lags

We prove a representation of the partial autocorrelation function α(・) of a stationary process { Xn : n ∈ Z}, in terms of the AR(∞) and MA(∞) coefficients. We apply it to show what α(n) looks like for large n, especially, when {Xn} is a long-memory process. For example, if {Xn} is a fractional ARIMA(p. d. q) process, then we have a(n)～d/n as n → ∞