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 real-world 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 specific 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 (q-th power of the absolute values of these) multiresolution 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 Brownian 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) covariance functions