 
Summary: On the impact of the number of vanishing
moments on the dependence structures of
compound Poisson motion and fractional
Brownian motion in multifractal time
B. Vedel, H. Wendt, P. Abry, S. Jaffard
Abstract From a theoretical perspective, scale invariance, or simply scaling, can
fruitfully be modeled with classes of multifractal stochastic processes, designed
from positive multiplicative martingales (or cascades). From a practical perspective,
scaling in realworld data is often analyzed by means of multiresolution quantities.
The present contribution focuses on three different types of such multiresolution
quantities, namely increment, wavelet and Leader coefficients, as well as on a spe
cific multifractal processes, referred to as Infinitely Divisible Motions and fractional
Brownian motion in multifractal time. It aims at studying, both analytically and by
numerical simulations, the impact of varying the number of vanishing moments of
the mother wavelet and the order of the increments on the decay rate of the (higher
order) covariance functions of the (qth power of the absolute values of these) mul
tiresolution coefficients. The key result obtained here consist of the fact that, though
it fastens the decay of the covariance functions, as is the case for fractional Brown
ian motions, increasing the number of vanishing moments of the mother wavelet or
the order of the increments does not induce any faster decay for the (higher order)
