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∂N ( s$_{1}$ ) and lim$_{h}$$_{↓}$$_{0}$ g$_{1}$$_{,h}$ ( t$_{0}$ ) = 1 / ‖ (1 , . . . , 1) ‖$_{D}$$_{1}$$_{,...,d}$ , but lim$_{h}$$_{↓}$$_{0}$ g$_{1}$$_{,h}$ ( t ) = 0 for all t / ∈ N ( s$_{1}$ ) due to Lemma 3.1. 

Convergence of the mean squared error. In this section we investigate a sequence of kernel-based generalized max-linear models, where the diameter of the grids decreases. We analyze under which conditions the integrated mean squared error of (ˆ η$_{t,h}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k converges to zero. We start with a general result on generator processes. 

Lemma 3.2. Let ( Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k be a generator of an SMSP and ε$_{n}$ , n ∈ N , be a null sequence. Then 

where ‖·‖ is an arbitrary norm on R $^{k}$. 

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Proof. The paths of ( Z$_{t}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k are continuous, so they are also uniformly contin- uous. Therefore, sup$_{‖}$$_{t}$$_{−}$$_{s}$$_{‖≤}$$_{ε}$$_{n}$ | Z$_{t}$ − Z$_{s}$ | →$_{n}$$_{→∞}$ 0. Furthermore, 

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with E ( sup$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k Z$_{t}$ ) < ∞ due to property (1) of a generator. The assertion now follows from the dominated convergence theorem. □ 

Let G$_{n}$ := { s$_{1}$$_{,n}$, . . . , s$_{d}$$_{(}$$_{n}$$_{)}$$_{,n}$ } , n ∈ N , be a set of distinct points in [0 , 1] k with the property 

where ε$_{n}$ →$_{n}$$_{→∞}$ 0. Define, for instance, G$_{n}$ in such a way that 

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with N ( s$_{i,n}$ ) as defined in (7). Clearly, d := d ( n ) →$_{n}$$_{→∞}$ ∞ . Denote by ‖·‖$_{D}$ ( n ) $^{s}$1,...,s d the D -norm pertaining to η$_{s}$$_{1}$$_{,n}$, . . . , η$_{s}$$_{d,n}$ . Let further ˆ η$_{n}$ = (ˆ η$_{t,n}$ )$_{t}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ k be the 

