  This paper addresses the problem of regularity properties of functions
represented as an expansion in a wavelet basis with random coefficients in
terms of finiteness of their Besov norm with probability 1. Such
representations are used to specify a prior measure in Bayesian nonparametric
wavelet regression. Investigating regularity of such functions is an important
problem since the support of the posterior measure does not include functions
that are not in the support of the prior measure, and therefore determines the
functions that are possible to estimate using specified Bayesian model. We
consider a more general parametrisation than has been studied previously which
allows to study a priori regularity of functions under a wider class of priors.
We also emphasise the difference between the abstract stochastic expansions
that have been studied in the literature, and the expansions actually arising
in nonparametric regression, and show that the latter cover a wider class of
functions than the former. We also extend these results to stochastic
expansions in an overcomplete wavelet dictionary.
