Matching of observations

TL;DR

This paper analyzes the statistical distribution of closest encounters in mixing dynamical systems, revealing a Gumbel distribution influenced by trajectory length, measure dimensions, and an extremal index, with implications for physical system analysis.

Contribution

It provides a new theoretical framework linking the distribution of closest encounters to generalized dimensions and extremal indices in dynamical systems.

Findings

Distribution is of Gumbel type for large trajectories

Distribution depends on trajectory length and measure dimensions

Extremal index formula provided for expanding maps

Abstract

We study the statistical distribution of the closest encounter between observations computed along different trajectories of a mixing dynamical system. At the limit of large trajectories, the distribution is of Gumbel type and depends on the length of the trajectories and on the Generalized Dimensions of the image measure. It is also modulated by an Extremal Index, for which we give a formula in the case of expanding maps of the interval and regular observations. We discuss the implications of these results for the study of physical systems.