Asymptotics for the partial autocorrelation function of a stationary process

Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes

“…So naturally we need an adequate Tauberian condition. Whereas we can use monotonicity as the necessary Tauberian condition in Inoue (2000), it is difficult to verify it in the present paper because we are lacking (C1). We overcome this trouble by verifying another Tauberian condition (Proposition 4.4) which is weaker than monotonicity but enough for our purpose.…”

“…Brockwell and Davis [(1991), §13.2])? We dealt with this specific problem in Inoue (2000) and showed that under appropriate conditions there exists a simple asymptotic formula for α (·). However, the main results of Inoue (2000) do not cover an important class of long-memory processes, that is, the fractional ARIMA (autoregressive integrated moving-average) model.…”

“…We recall the results of Inoue (2000) that are closely related to Theorem 1.1. Let −∞ < d < 1/2 and (·) be a slowly varying function at infinity (cf.…”

“…1]). Then Theorem 2.1 of Inoue (2000) shows that, under certain conditions on the MA(∞) coefficients c n and the AR(∞) coefficients a n (see §2) of a stationary process {X n },…”

Prediction of long memory processes on same-realisation

“…Assumption H.2 does not imply the bound on the coefficients (a j ) j∈N and (b j ) j∈N given in assumption H.1. Inoue (2000) has proved that the asymptotic expression of the autocovariance σ(k) ∼ L(k)k 2d−1 implies:…”