Whittle estimator for finite-variance non-Gaussian time series with long memory

We consider time series Yₜ = G(Xₜ) where Xₜ is Gaussian with long memory and G is a polynomial. The series Yₜ may or may not have long memory. The spectral density g_θ(x) of Yₜ is parameterized by a vector θ and we want to estimate its true value θ₀ . We use a least-squares Whittle-type estimator θ̂N for θ₀, based on observations Y₁,,YN. If Yₜ is Gaussian, then √N(θ̂N-θ₀) converges to a Gaussian distribution. We show that for non-Gaussian time series Yₜ , this √N consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if Yₜ has short memory.