Parameter estimation for a misspecified ARMA model with infinite variance innovations

We consider a general linear model X_t = ∑ _j=-∞^∞ ψ_jZ_t-j, where the innovations Z_t belong to the domain of attraction of an α-stable law for α < 2, so that neither Z_t nor X _t have a finite variance. We do not assume that (X_t) is a standard ARMA process of the form φ(B)X_t = θ(B)Z _t, but we fit an ARMA process of a given order to the data X _l,..., X_n by estimating the coefficients of φ and θ. Given that (X_t) is an ARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of θ and φ. Moreover, it then has a heavy-tailed limit distribution and the rate of convergence is (n/log n)^1/α, which compares favorably with the L² situation with rate √n. In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily an ARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution.