Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

Abstract

In this paper we study the distribution function $P(u_α)$ of the estimators $u_α \sim T^{-1} \int^T_0 \, ω(t) \, {\bf B}^2_{t} \, dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\bf B}_{t}$. We pursue here the optimisation further by considering a family of weight functions of the form $ω(t) = (t_0 + t)^{-α}$, where $t_0$ is a time lag and $α$ is an arbitrary real number, and seeking such values of $α$ for which the estimators most efficiently filter out the fluctuations. We calculate $P(u_α)$ exactly for arbitrary $α$ and arbitrary spatial dimension $d$, and show that only for $α= 2$ the distribution $P(u_α)$ converges, as $ε= t_0/T \to 0$, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with $α= 2$ possess an ergodic property, so that the ensemble averaged diffusion coefficient can be obtained with any necessary precision from a single trajectory data, but at the expense of a progressively higher experimental resolution. For any $α\neq 2$ the distribution attains, as $ε\to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.

27 pages, 5 figures