 Variance of a long-time mean in a harmonic trap

For stationary covariance $C(t)=(k_{B}T/k)e^{-|t|/{\tau }_k}$, the mean over duration $\mathcal{T}$ has variance $2(k_{B}T/k)[{\tau }_k/\mathcal{T}-({\tau }_k/\mathcal{T}{)}^2(1-e^{-\mathcal{T}/{\tau }_k})]$. At $\mathcal{T}\gg {\tau }_k$ this becomes $2(k_{B}T/k){\tau }_k/\mathcal{T}$. Closely sampled positions remain correlated and cannot be counted as independent.