Cross-correlation of long-range correlated series

TL;DR

This paper introduces a method to estimate cross-correlation in long-range correlated series, applicable to diverse fields like finance and genomics, revealing scale-dependent relationships and stationarity properties.

Contribution

A novel approach for estimating cross-correlation functions of long-range correlated series across different scales and lags, with theoretical and practical applications.

Findings

Cross-correlation depends only on lag for fractional Brownian motions.

The cross-correlation scales as a power of the scale parameter with exponent H1+H2.

Application to financial and genomic data demonstrates method's utility.

Abstract

A method for estimating the cross-correlation $C_{xy}(\tau)$ of long-range correlated series $x(t)$ and $y(t)$, at varying lags $\tau$ and scales $n$, is proposed. For fractional Brownian motions with Hurst exponents $H_1$ and $H_2$, the asymptotic expression of $C_{xy}(\tau)$ depends only on the lag $\tau$ (wide-sense stationarity) and scales as a power of $n$ with exponent ${H_1+H_2}$ for $\tau\to 0$. The method is illustrated on (i) financial series, to show the leverage effect; (ii) genomic sequences, to estimate the correlations between structural parameters along the chromosomes.