Kernel estimation for quadratic functional of long memory linear processes with infinite variance

Let X = {X-n : n is an element of 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 estimate the quadratic functional integral(R) f(2)(x) dx of the linear process X by using the kernel estimator T-n(h(n)) = 2/n(n - 1)h(n) Sigma(1 <= j<i <= n) K(X-i - X-j/h(n)). Moreover, using the Delta method, we obtain the corresponding results for the kernel estimator of the quadratic Renyi entropy - ln(integral(R) f(2)(x) dx). When innovations are symmetric alpha-stable random variables, we give the simulation study for these two kernel estimators.