Testing the time series for multifractality

Multifractality has become a central concept for characterizing complexity across scientific domains. Yet, its reliable detection in empirical time series remains challenging due to the interplay of temporal correlations and heavy-tailed fluctuations. Recent analytical and numerical results demonstrate that genuine multifractality cannot arise in the absence of long-range temporal correlations. In contrast, apparent multifractal spectra observed in shuffled or short uncorrelated data are finite-size artifacts or manifestations of bifractality in Lévy-stable regimes. Motivated by these findings, we revisit multifractal detrended fluctuation analysis (MFDFA) as the most stable and widely used tool for testing multiscaling properties in complex systems, from financial markets to natural language. We show, using controlled multiplicative cascades, how the distribution of fluctuations—modulated via q-Gaussian transformations—can broaden the singularity spectrum only when correlations are already present. This provides a principled way to separate true multifractal effects from those induced by fatter tails. We further demonstrate that empirical systems often exhibit multiscale organization consistent with correlation-driven multifractality, including financial markets, where nonlinear dependencies and volatility clustering contribute to the observed spectra. Additional analogies arise in linguistic time series, where long-range dependencies and hierarchical structure produce fractal or multifractal patterns even in symbolic data. Building on these theoretical and empirical insights, we propose a robust protocol for testing multifractality that incorporates surrogate data, tail-stabilized reference distributions, and scale-range diagnostics. This procedure allows researchers to quantify how much of the spectrum width originates from temporal organization versus the heaviness of fluctuations. The talk concludes by emphasizing that properly disentangling these components is essential for interpreting multifractality as a genuine marker of complexity in natural and socio-economic systems.