Kernel entropy estimation for long memory linear processes with infinite variance

Abstract

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process with innovations in the domain of attraction of an $\alpha$-stable law $(0<\alpha<2)$. Assume that the linear process $X$ has a bounded probability density function $f(x)$. Then, under certain conditions, we consider the estimation of the quadratic functional $\int_{\mathbb{R}} f^2(x) \,dx$ by using the kernel estimator \[ T_n(h_n)=\frac{2}{n(n-1)h_n}\sum_{1\leq j<i\leq n}K\left(\frac{X_i-X_j}{h_n}\right). \] The simulation study for long memory linear processes with symmetric $\alpha$-stable innovations is also given.