Sampling uncertainty of an endpoint diffusivity estimate ($3n\hat{D}/D\sim {\chi }_{3n}^2$)

Each Cartesian Brownian increment has variance $2Dt$. Summing their normalized squares gives a chi-squared distribution with $3n$ degrees of freedom. Hence $\hat{D}$ is unbiased and has relative standard deviation $\sqrt{2/(3n)}$. Its inverse is not unbiased: for $3n>2$, $\mathbb{E}[1/\hat{D}]=[3n/(3n-2)]/D$. Thus the direct inverse-diffusivity radius estimate has appreciable finite-sample bias when only a few trajectories are observed. Measurement noise or drift adds separate systematic error.