functions. 

The theoretical results of this paper extend the usual representation of the density, conditional distribution and density and conditional moments to situations where these may not exist in an ordinary sense. The advantage that this approach provides is its generality. On the other hand, the topology in the spaces of generalized functions is weak and well-posedness does not imply convergence in norm. 

The asymptotic results provide a general approach, so that when the usual assumptions may fail there is still a sense in which consistency holds. Moreover a root-n convergence rate obtains, again as a consequence of the weak topology with no guarantee of good convergence in norm. The practical advantage is in the possibility of utilizing the generalized random process and its limit process for inference without making any restrictions on the distribution. 

## 2 Density as solution to a well-posed inverse problem in the space of generalized func- tions 

For the definitions and results pertaining to spaces of generalized functions the main references are to books by Schwartz (1966) Gel’fand and Shilov (1964). A useful summary is in Zinde-Walsh (2008, 2012); the main defini- 

