b, F t ( ψ$_{v}$ ))) →$_{p}$ 0 . Similarly for F t − $^{1}$. (b) If µ ∈ G and b$_{n}$ − b →$_{p}$ 0 in G $^{∗}$, then b$_{n}$µ − bµ →$_{p}$ 0 in G $^{∗}$. This follows similarly from the fact that µψ ∈ G for ψ ∈ G. (c) Parts (a) and (b) of this Remark also hold with →$_{a.s.}$ replacing →$_{p}$ and with convergence to zero replaced by convergence in distribution to a limit generalized random process. Suppose that the known functions, w or w$_{1}$, w$_{2}$$_{k}$ (equivalently, ε or ε$_{1}$, ε$_{2}$$_{k}$ ) are consistently estimated in S $^{∗}$. In the models discussed here these func- tions are density functions or conditional mean functions; typically these are ordinary functions for which commonly used nonparametric estimators are shown to be consistent pointwise or in some norm (uniform or L$_{1}$, say) under some sufficient conditions. If the functions and the estimators can be repre- sented as elements in S ∗ then convergence in many common norms implies convergence in the weaker topology of S $^{∗}$, so that such consistent estimators are consistent in S ∗ and the following discussion of consistency of the solu- tions to convolution equations considered here applies. However, consistency in S ∗ applies more widely than the usual consistency results. For exam- ple, as shown in Zinde-Walsh, 2008, kernel estimators of density converge as stochastic generalized functions even when they diverge pointwise (e.g. at 5 Consistent estimation of solutions to stochas- tic equations 48