For example, if as in Cunha et al (2010) both γ and φ are characteristic functions and thus continuous and bounded, products exist and Theorem 5(b) applies, as long as the conditions of Theorem 3 are satisfied and the condition (21) is satisfied for the inverses of the functions in the sequence, that is e.g. under Theorem 3(a) for γ − 1 n , or under 3(b) for φ − 1 n , so that the corresponding distributions are not supersmooth and the polynomial lower bound holds uniformly, well-posedness holds. 

The implications that well-posedness has for estimation are two-fold. One is that unless well-posedness holds, that is the inverse mapping from the class of the known functions into the class of identified solutions is continuous, the solutions corresponding to the consistently estimated known functions will not in general provide consistent estimators for the solutions. The other is that in a well-posed problem consistent estimation of the known functions automatically gives rise to consistency of plug-in estimators of solutions; the next section examines consistent estimation of the deconvolution and the solution to the system (3) in the space S $^{∗}$. 

## 4 Random generalized functions and stochas- tic convergence 

This section examines stochastic convergence of the solutions to the decon- volution equation (2) and to (3), equivalently to (15) and (16) . If some 

