Stochastic Expansions in an Overcomplete Wavelet Dictionary

We consider random functions defined in terms of members of an overcomplete wavelet dictionary. The function is modelled as a sum of wavelet components at arbitrary positions and scales where the locations of the wavelet components and the magnitudes of their coefficients are chosen with respect to a marked Poisson process model. The relationships between the parameters of the model and the parameters of those Besov spaces within which realizations will fall are investigated. The models allow functions with specified regularity properties to be generated. They can potentially be used as priors in a Bayesian approach to curve estimation, extending current standard wavelet methods to be free from the dyadic positions and scales of the basis functions.