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

We consider time series $Y_t = G(X_t)$ where $X_t$ is Gaussian with long memory and $G$ is a polynomial. The series $Y_t$ may or may not have long memory. The spectral density $g_\theta(x)$ of $Y_t$ is parameterized by a vector $\theta$ and we want to estimate its true value $\theta_0$ . We use a least-squares Whittle-type estimator $\hat{\theta}_N$ for $\theta_0$, based on observations $Y_1,\dots,Y_N$. If $Y_t$ is Gaussian, then $\sqrt{N}(\hat{\theta}_N-\theta_0)$ converges to a Gaussian distribution. We show that for non-Gaussian time series $Y_t$ , this $\sqrt{N}$ consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if $Y_t$ has short memory.