Kernel entropy estimation for long memory linear processes with infinite variance

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 $$T_n(h_n)=\frac{2}{n(n-1)h_n}\sum _{1\le j<i\le n}K\left(\frac{X_i-X_j}{h_n}\right)\mathrm{.}$$ The simulation study for long memory linear processes with symmetric $\alpha$-stable innovations is also given.