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

Abstract

In this paper we study the distribution function $P(u_{\alpha })$ of the estimators $u_{\alpha }\sim T^{-1}{\int }_0^T\omega (t){\mathbf{B}}_t^{2}dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\mathbf{B}}_t$. 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 $\alpha \ne 2$ the distribution attains, as $\epsilon \to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.

27 pages, 5 figures

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