= Gaussian autoregressive kernel variance correction
{c}
{title2=$g_h(j)=\sqrt\pi[(1+h^2-\rho^j)^{-1/2}-(1+h^2)^{-1/2}]$}
For a stationary <autoregressive process of order one> with unit-variance <normal distribution> marginals and fixed $0<\rho<1$, the <Gaussian kernel> covariance correction at lag $j$ has the displayed form. The difference of two observations has variance $2(1-\rho^j)$, and their <characteristic functions> reduce the Fourier integral to the difference of two <Gaussian integrals>. The <mean value theorem> gives $0\leq g_h(j)\leq\sqrt\pi\rho^j/[2(1-\rho)^{3/2}]$, uniformly over $h>0$. Thus the integrated variance differs from its independent-sample counterpart by $O(n^{-1})$. The marginal bias is unchanged, so this dependence does not alter the leading optimal bandwidth or the $n^{-4/5}$ <mean integrated squared error> for a second-order density kernel.
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