On joint profile multifractal formalisms

In this paper, we formulate joint profile formalisms based on statistics of histograms of joint multiscale quantities. These formalisms allow the simultaneous multifractal analysis of several pointwise exponents derived from collections of measures, functions, signals, or images. We first show that joint profiles improve the upper bounds for joint spectra of regularities that can be derived from joint Legendre spectra, and have the advantage to recover non-concave joint spectra. We illustrate these facts for some bivariate scale-lacunary wavelet series. These series are very important in applications since they mimic the typical behavior of sequences of independent random variables better suited to real-world data modeling. We also show the importance of synchronicity. Then, we focus on the setting of joint $ p_i $-wavelet leaders, $ 1 \leq i \leq I $, $ I \in \mathbb{N} $. We prove that the upper bounds are optimal since they become equalities for some multivariate functions. Finally, we prove generic related results in the sense of Baire's categories.