The goal of multifractal analysis is to characterize the variations in local regularity of functions or signals by computing the Hausdorff dimension of the sets of points that share the same regularity. While classical approaches rely on Hölder exponents and are limited to locally bounded functions, the notion of $p$-exponents extends multifractal analysis to functions locally in $L^p$, allowing a rigorous characterization of singularities in more general settings. In this work, we propose a wavelet-based methodology to estimate the $p$-spectrum from the distribution of wavelet coefficients across scales. First, we establish an upper bound for the $p$-spectrum in terms of this distribution, generalizing the classical Hölder case. The sharpness of this bound is demonstrated for \textit{Random Wavelet Series}, showing that it can be attained for a broad class of admissible distributions of wavelet coefficients. Finally, within the class of functions sharing a prescribed wavelet statistic, we prove that this upper bound is realized by a prevalent set of functions, highlighting both its theoretical optimality and its representativity of the typical multifractal behaviour in constrained function spaces.

We prove a convergence result for a large class of random models that encompasses the case of the BPHZ models used in the study of singular stochastic PDEs. We introduce for that purpose a useful variation on the notion of regularity structure called a regularity-integrability structure. It allows to deal in a single elementary setting with models on a usual regularity structure and their first order Malliavin derivative.