Kernel entropy estimation for linear processes

Let $\{X_n: n\in \mathbb{N}\}$ be a linear process with bounded probability density function $f(x)$. We study the estimation of the quadratic functional $\int_{\mathbb{R}} f^2(x)\, dx$. With a Fourier transform on the kernel function and the projection method, it is shown that, under certain mild conditions, the estimator $$\frac{2}{n(n-1)h_n} \sum_{1\le i<j\le n}K\left(\frac{X_i-X_j}{h_n}\right)$$ has similar asymptotical properties as the i.i.d. case studied in Gin\'{e} and Nickl (2008) if the linear process $\{X_n: n\in \mathbb{N}\}$ has the defined short range dependence. We also provide an application to $L^2_2$ divergence and the extension to multivariate linear processes. The simulation study for linear processes with Gaussian and $\alpha$-stable innovations confirms our theoretical results. As an illustration, we estimate the $L^2_2$ divergences among the density functions of average annual river flows for four rivers and obtain promising results.