Characterization of p-exponents by continuous wavelet transforms, applications to the multifractal analysis of sum of random pulses

Abstract: The theory of orthonormal wavelet bases is a useful tool in multifractal analysis, as it provides a characterization of the different exponents of pointwise regularities (H{\"o}lder, p-exponent, lacunarity, oscillation, etc.). However, for some homogeneous self-similar processes, such as sums of random pulses (sums of regular, well-localized functions whose expansions and translations are random), it is easier to estimate the spectrum using continuous wavelet transforms. In this article, we present a new characterization of p-exponents by continuous wavelet transforms and we provide an application to the regularity analysis of sums of random pulses.