A note on power-law cross-correlated processes

Abstract

Abstract In this paper, some mathematical support is provided to properly justify the validity of the so-called multifractal height cross-correlation analysis (MFHXA), first contributed by Kristoufek (2011)[1]. With this aim, we extend several concepts from univariate random functions and their increments to the bivariate case. Specifically, we introduce the bivariate cumulative range as well as the concepts of co-self-similar processes and random processes with co-affine increments of bivariate parameter. They allow us to introduce a new procedure, named multifractal cross-correlated fractal dimension (MFXFD) algorithm. We theoretically prove the validity of such a novel approach to calculate the bivariate Hurst exponent of a pair of processes with co-affine increments. Interestingly, that class of random functions is characterised theoretically in terms of co-self-similar processes. Moreover, we prove that a pair of univariate self-similar processes are co-self-similar and their bivariate Hurst exponent coincides with the mean of their univariate parameters. Finally, we test the behavior of the new algorithm to calculate the bivariate Hurst exponent of a pair of time series. Both DCCA and MFHXA procedures are involved in such an empirical comparison. Our results suggest that the new MFXFD performs (at least) as well as the other approaches with shorter deviations with respect to the mean of the bivariate Hurst exponents.