Multifractal behavior and causality effects in the time series of the blazar Mrk 421

This study investigates the multifractal structure of time series of the blazar Mrk 421, across different energy bands, with a focus on the analysis of the scaling exponents and power spectral density (PSD) within the framework of random fractal theory. We calculated fractional variability (Fvar) for comparison with the results obtained from multifractal measures derived from the Multifractal Detrended Moving Average (MFDMA) analysis, which allows for a detailed examination of the variability characteristics observed in quasar emissions in various energy bands. A Granger causality test reveals significant evidence that the lower-energy series can predict the higher-energy series, indicating that variations in the soft X-ray band precede those in the hard X-ray band. This behaviour is consistent with energy-dependent particle acceleration processes in a jet-dominated scenario, where lower-energy synchrotron emission is produced first, followed by higher-energy emission from relativistic electrons. The transfer entropy analysis corroborates the Granger causality results, providing additional evidence of directional information flow between energy bands and uncovering possible external drivers influencing the observed dynamics. Our multifractal analysis further supports these interpretations, with the higher energy band exhibiting a multifractality degree twice that of the soft band, suggesting a more intricate and dynamic emission process.