consistent estimators are available for either the function w ( w$_{1}$ and w$_{2}$$_{k}$ ) or, equivalently, for the Fourier transform, ε ( ε$_{1}$ and ε$_{2}$$_{k}$ ) , stochastic conver- gence of the solutions provides consistency results for plug-in estimators of g as long as well-posedness holds. Since generalized functions in any space G ∗ are represented as linear continuous functionals on G results for random functionals are applicable here, the specific feature is that for any generalized function b ∈ G ∗ one needs to consider the collection of random functionals indexed by the functions from the space G. Following Gel’fand and Vilenkin (1964) define random generalized functions as random linear continuous functionals on the space of test functions (see e.g. Koralov and Sinai, 2007 who consider specifically S $^{∗}$− ch.17). In particular, any random generalized function ˜ b on G is represented by a collection of (complex-valued) random variables on a common probability space that are indexed by ψ ∈ G, denoted ( ˜ b, ψ ) , such that 4.1 Random generalized functions (a) ( ˜ b, ( a$_{1}$ψ$_{1}$ + a$_{2}$ψ$_{2}$ )) = a$_{1}$ ( ˜ b, ψ$_{1}$ ) + a$_{2}$ ( ˜ b, ψ$_{2}$ ) a.s.; (b) if ψ$_{kn}$ → ψ$_{k}$ in S as n → ∞ , k = 1 , 2 .., m, then vectors (( ˜ b, ψ$_{1}$$_{n}$ ) ... ( ˜ b, ψ$_{mn}$ )) →$_{d}$ (( ˜ b, ψ$_{1}$ ) ... ( ˜ b, φ$_{m}$ )) where →$_{d}$ denotes convergence in distribution. As shown in Gel’fand and Vilenkin, equivalently, there exists a probabil- 45