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.

本文提供了一些数学支持，以正确证明所谓的多重分形高度互相关分析（MFHXA）的有效性，该模型首先由Kristoufek（2011）提出[1]。为此，我们将几个概念从单变量随机函数及其增量扩展到双变量情况。具体来说，我们介绍了双变量累积范围以及具有双变量参数的仿射增量的自相似过程和随机过程的概念。它们使我们可以引入一个新的过程，称为多重分形互相关分形维数（MFXFD）算法。我们从理论上证明了这种新颖方法用于计算具有协仿射增量的一对过程的二元Hurst指数的有效性。有趣的是 该类随机函数在理论上是根据共同相似过程来表征的。此外，我们证明了一对单变量自相似过程是共同自相似的，并且它们的双变量Hurst指数与它们的单变量参数的平均值一致。最后，我们测试新算法的行为，以计算一对时间序列的二元Hurst指数。DCCA和MFHXA程序都涉及这种经验比较。我们的结果表明，新的MFXFD至少与双变量Hurst指数的均值相比具有（至少）以及其他方法具有更短的偏差。最后，我们测试新算法的行为，以计算一对时间序列的二元Hurst指数。DCCA和MFHXA程序都涉及这种经验比较。我们的结果表明，新的MFXFD至少与双变量Hurst指数的均值相比具有（至少）以及其他方法具有更短的偏差。最后，我们测试新算法的行为，以计算一对时间序列的二元Hurst指数。DCCA和MFHXA程序都涉及这种经验比较。我们的结果表明，新的MFXFD至少与双变量Hurst指数的均值相比具有（至少）以及其他方法具有更短的偏差。