Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Distribution of the least-squares estimators of a single Brownian
  trajectory diffusion coefficient   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.

29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom