Abstract:
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Let $X_n=\sum\limits_{i=0}^\infty a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are i.i.d. centered random variables taking values in $\R^d$ and $\sum\limits^{\infty}_{i=0}|a_i|< \infty$. Assume that $f$ is the probability density function of $X_n$. We consider the estimation of the quadratic functional $\int_{\R^d} f^2(x)\, dx$. It is shown that, under certain conditions, the estimator \[ \frac{2}{n(n-1)h^d_n} \sum_{1\le i< j\le n}K\left(\frac{X_i-X_j}{h_n}\right) \] is asymptotically efficient. For i.i.d. case, Gin\'{e} and Nickl applied a convolution method to obtain the bias, and used the Hoeffding's decomposition for $U$-statistics to study the stochastic part. Instead, for linear processes, we apply the Fourier transform on the kernel function to derive the asymptotic properties. The result and the method have application to $L_2^2$ divergence between distributions of two linear processes.
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