Cross-correlation of long-range correlated series

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.