8 Conclusion and further questions The approach employed here makes it possible to avoid any restrictions when defining density, conditional distribution and conditional density as well as conditional moments for a smooth function (e.g. conditional expectation or second moment). The usual kernel estimators converge to the limit generalized functions at a parametric rate; the limit process is provided by a Gaussian process in the space of generalized functions, that is a Gaussian process indexed by well-behaved functions from the appropriate spaces. The results here were based on a random sample of observations to sim- plify exposition; extension to stationary ergodic or mixing processes can be obtained. Further extensions to relax homogeneity and independence are a subject of future research. The limit results imply that with a judicial selection of indexing functions one could use the kernel estimators for inference in very general situations; this investigation is mostly left for future research. 9 Appendix. Proof of Theorem 2. Define a generalized function e$_{nhj}$ such that the value of the functional 32