14 MICHAEL FALK, MAXIMILIAN ZOTT kernel-based discretized version of η with grid G$_{n}$ , that is, where for i = 1 , . . . , d K : [0 , ∞ ) → [0 , 1] is the continuous and strictly decreasing kernel function sat- isfying condition (6) and h$_{n}$ , n ∈ N , is some positive sequence. We have already seen in Lemma 3.1 that g$_{i,n}$ ( t ) ∈ [0 , 1], t ∈ [0 , 1] $^{k}$, n ∈ N . Furthermore we have the following result. Lemma 3.3. Choose t ∈ [0 , 1] $^{k}$. There is a sequence i ( n ) , n ∈ N , such that t ∈ ⋂ n ∈ $_{N}$N ( s$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ ) . Define g$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ and ε$_{n}$ as above, n ∈ N . Then if ε$_{n}$ →$_{n}$$_{→∞}$ 0 , h$_{n}$ →$_{n}$$_{→∞}$ 0 , ε$_{n}$/h$_{n}$ →$_{n}$$_{→∞}$ ∞ . Proof. Let t ∈ [0 , 1] k and choose a sequence i ( n ), n ∈ N , as above. Put for simplicity s$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ =: s$_{i,n}$ and g$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ =: g$_{i,n}$ . We have From t ∈ N ( s$_{i,n}$ ) we conclude ‖ t − s$_{i,n}$ ‖ ≤ ε$_{n}$ . Hence, we have due to (1) and the properties of the kernel function K ˆ η$_{t,n}$ = max i =1 ,...,d η$_{s}$$_{i,n}$ g$_{i,n}$ ( t ) , t ∈ [0 , 1] $^{k}$, g$_{i,n}$ ( t ) = K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) ‖ ( K ( ‖ t − s$_{1}$$_{,n}$ ‖ /h$_{n}$ ) , . . . , K ( ‖ t − s$_{d,n}$ ‖ /h$_{n}$ )) ‖$_{D}$ ( n ) $^{s}$1,...,s d , t ∈ [0 , 1] $^{k}$, lim n $_{→∞}$g$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ ( t ) = 1 , 1 ≥ g$_{i,n}$ ( t ) = K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) E ( max$_{j}$$_{=1}$$_{,...,d}$ K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) Z$_{s}$$_{j,n}$ ) ≥ ( E ( max$_{j}$$_{:}$$_{‖}$$_{s}$$_{j,n}$$_{−}$$_{t}$$_{‖≥}$$_{2}$$_{ε}$$_{n}$ K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) Z$_{s}$$_{j,n}$ ) K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) + E ( max$_{j}$$_{:}$$_{‖}$$_{s}$$_{j,n}$$_{−}$$_{t}$$_{‖}$$_{<}$$_{2}$$_{ε}$$_{n}$ K ( ‖ t − s$_{j,n}$ ‖ /h$_{n}$ ) Z$_{s}$$_{j,n}$ ) K ( ‖ t − s$_{i,n}$ ‖ /h$_{n}$ ) ) − 1 =: ( A$_{i,n}$ ( t ) + B$_{i,n}$ ( t )) − $^{1}$. 0 ≤ A$_{i,n}$ ( t ) ≤ K (2 ε$_{n}$/h$_{n}$ ) K ( ε$_{n}$/h$_{n}$ ) E ( sup t ∈ [0 , 1] k Z$_{t}$ ) →$_{n}$$_{→∞}$ 0 ,