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

TL;DR

This paper analyzes the distribution of least-squares estimators for a diffusion coefficient in Brownian motion, identifying the optimal weight function for ergodic estimation from single trajectories.

Contribution

It provides an exact calculation of the estimator distribution for various weight functions and identifies the specific weight that ensures ergodicity in diffusion coefficient estimation.

Findings

Only for α=2 does the estimator distribution converge to a delta function.

Estimators with α=2 are ergodic and can accurately estimate the diffusion coefficient from a single trajectory.

Other estimators have finite variance and are non-ergodic.

Abstract

In this paper we study the distribution function $P(u_{\alpha})$ of the estimators $u_{\alpha} \sim T^{-1} \int^T_0 \, \omega(t) \, {\bf B}^2_{t} \, dt$, 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 $\omega(t) = (t_0 + t)^{-\alpha}$, where $t_0$ is a time lag and $\alpha$ is an arbitrary real number, and seeking such values of $\alpha$ for which the estimators most efficiently filter out the fluctuations. We calculate $P(u_{\alpha})$ exactly for arbitrary $\alpha$ and arbitrary spatial dimension $d$, and show that only for $\alpha = 2$ the distribution $P(u_{\alpha})$ converges, as $\epsilon = t_0/T \to 0$, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with $\alpha = 2$ 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 $\alpha \neq 2$ the distribution attains, as $\epsilon \to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.