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

hal.structure.identifierInstituto de Fisica
hal.structure.identifierCentro de Ciencias de la Complejidad
dc.contributor.authorBOYER, Denis
hal.structure.identifierLaboratoire Ondes et Matière d'Aquitaine [LOMA]
dc.contributor.authorDEAN, David S.
hal.structure.identifierLaboratory of Physical Properties
hal.structure.identifierDepartment of Mathematics and Statistics [Helsinki]
dc.contributor.authorMEJIA-MONASTERIO, Carlos
hal.structure.identifierLaboratoire de Physique Théorique de la Matière Condensée [LPTMC]
dc.contributor.authorOSHANIN, Gleb
dc.date.created2013-01-18
dc.date.issued2013
dc.identifier.issn1742-5468
dc.description.abstractEnIn 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.
dc.language.isoen
dc.publisherIOP Publishing
dc.rights.urihttp://creativecommons.org/licenses/by-nc/
dc.subject.enBrownian motion
dc.subject.endata mining (theory)
dc.subject.ensingle molecule
dc.subject.endiffusion
dc.title.enDistribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient
dc.typeArticle de revue
dc.identifier.doi10.1088/1742-5468/2013/04/P04017
dc.subject.halPhysique [physics]/Matière Condensée [cond-mat]/Mécanique statistique [cond-mat.stat-mech]
dc.identifier.arxiv1301.4374
bordeaux.journalJournal of Statistical Mechanics: Theory and Experiment
bordeaux.pageP04017 (1-24)
bordeaux.volume2013
bordeaux.issue4
bordeaux.peerReviewedoui
hal.identifierhal-00825399
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00825399v1
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