Complex systems are composed of mutually interacting components and the output values of these components usually exhibit long-range cross-correlations. Using wavelet analysis, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call multifractal cross wavelet analysis (MFXWT). We assess the performance of the MFXWT method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. For binomial multifractal measures, we find the empirical joint multifractality of MFXWT to be in approximate agreement with the theoretical formula. For bFBMs, MFXWT may provide spurious multifractality because of the wide spanning range of the multifractal spectrum. We also apply the MFXWT method to stock market indices, and in pairs of index returns and volatilities we find an intriguing joint multifractal behavior. The tests on surrogate series also reveal that the cross correlation behavior, particularly the cross correlation with zero lag, is the main origin of cross multifractality.

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The behaviors of fat-tailed distribution, linear long memory, and nonlinear long memory are considered as possible sources of apparent multifractality. Which behavior should be preserved in null models plays an important role in statistical tests of empirical multifractality. In this paper, we compare the performance of two null models on testing the existence of multifractality in fractional Brownian motions (fBm), Markov-switching multifractal (MSM) model, and financial returns. One null model is obtained by shuffling the original data, which keeps the distribution unchanged. The other null model is generated by the iterative amplitude adjusted Fourier transform (IAAFT) algorithm, which insures that the surrogate data and the original data sharing the same distribution and linear long memory behavior. We find that the tests based on the shuffle null model only reject the multifractality in fBm with H = 0.5 and the tests based on the IAAFT null model reject the multifractality in fBms (except for H = 0.1). And the multifractality in MSM and financial returns are significantly supported by the tests based on both null models. Our findings also shed light on the necessity of choosing suitable null models to test multifractality in other complex systems.

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