# Covariance of the running range of a Brownian trajectory

## TL;DR

This paper derives the exact covariance function of the running range of Brownian motion over two time intervals, revealing complex correlations between extremal values at different times.

## Contribution

It provides the first exact calculation of the covariance of the running range of Brownian motion and analyzes its asymptotic behavior.

## Key findings

- Exact covariance function derived for the running range.
- Non-trivial correlations between extremal values on different intervals.
- Asymptotic analysis of the covariance function.

## Abstract

The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a Brownian Bridge and of a Slepian process. Here, we focus on the two-time correlations of the running range of Brownian motion - the maximal extent of a Brownian trajectory on a finite time interval. We calculate exactly the covariance function of the running range and analyse its asymptotic behaviour. Our analysis reveals non-trivial correlations between the value of the largest descent (rise) of a BM from the top to a bottom on some time interval, and the value of this property on a larger time interval.