Variance of a long-time mean in a harmonic trap
= Variance of a long-time mean in a harmonic trap
For stationary covariance $C(t)=(k_BT/k)e^{-|t|/\tau_k}$, the <mean> over duration $\mathcal T$ has <variance> $2(k_BT/k)[\tau_k/\mathcal T-(\tau_k/\mathcal T)^2(1-e^{-\mathcal T/\tau_k})]$. At $\mathcal T\gg\tau_k$ this becomes $2(k_BT/k)\tau_k/\mathcal T$. Closely sampled positions remain correlated and cannot be counted as independent.