 Gaussian autoregressive kernel variance correction

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 $0\le g_h(j)\le \sqrt{\pi }{\rho }^j/[2(1-\rho {)}^{3/2}]$, 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.