Assume that the \({\mathbb{R}}^ r\times {\mathbb{R}}^ s\) valued random vector (X,Y) has a conditional density f(y\(| x)\) for Y given \(X=x\), \(x\in {\mathbb{R}}^ r\). The authors study a kernel estimator for f(y\(| x)\) based on an i.i.d. sample \((X_ j,Y_ j)^ n_{j=1}\), namely \[ \hat f_ n(y| x)=\sum^{n}_{j=1}K_ 1((X_ j-x)/h) K_ 2((Y_ j- v)/h)\cdot h^{-s}/\sum^{n}_{\nu =1}K_ 1((X_{\nu}-x)/h), \] where \(h=h(n)>0\) is a null-sequence and \(K_ 1\) and \(K_ 2\) are densities on \({\mathbb{R}}^ r\) and \({\mathbb{R}}^ s\) resp.. This estimator was also studied by D. Sun [Chin. J. Appl. Probab. Stat. 1, 95-102 (1985; Zbl 0583.62034)].
Now suppose that (essentially) \(K_ 1(v)=c_ 1\chi (\| v\| <\rho_ 1)\) and \(0\leq K_ 2(v)\leq c_ 2\chi (\| v\| <\rho_ 2)\) with positive constants \(c_ 1,c_ 2,\rho_ 1,\rho_ 2\) where \(\| \cdot \|\) denotes the Euclidean - or sup-norm. It is proved that \[ \lim_{n\to \infty}\hat f_ n(y| x)=f(y| x)\quad a.s. \] for \(F\times \lambda\) almost all (x,y), provided that \(nh^{r+s}/\log n\to \infty\). Hereby F denotes the distribution of X and \(\lambda\) is the Lebesgue-measure on \({\mathbb{R}}^ s\). The analytic key to the proof is a result on relative differentiation of measures (Lemma 1). A related approach was used by L. Devroye [Ann. Stat. 9, 1310-1319 (1981; Zbl 0477.62025)] in the context of nonparametric regression.