content_image/304146.jpg content_image/304145.jpg content_image/304148.jpg content_image/304149.jpg content_image/304147.jpg
since ε$_{n}$/h$_{n}$ →$_{n}$$_{→∞}$ ∞ by assumption. Furthermore, t ∈ N ( s$_{i,n}$ ) and the fact that K is decreasing implies 

Thus, 

<img src='content_image/304146.jpg'>

<img src='content_image/304145.jpg'>

because of Lemma 3.2. Note that ‖ s$_{j,n}$ − t ‖ < 2 ε$_{n}$ and t ∈ N ( s$_{i,n}$ ) imply ‖ s$_{j,n}$ − s$_{i,n}$ ‖ < 3 ε$_{n}$ . □ 

We have now gathered the tools to prove convergence of the mean squared error to zero. 

Theorem 3.4. Define ˆ η$_{n}$ and ε$_{n}$ as above, n ∈ N . Then for every t ∈ [0 , 1] k 

and 

if ε$_{n}$ →$_{n}$$_{→∞}$ 0 , h$_{n}$ →$_{n}$$_{→∞}$ 0 , ε$_{n}$/h$_{n}$ →$_{n}$$_{→∞}$ ∞ . 

Proof. Denote by 

<img src='content_image/304148.jpg'>

<img src='content_image/304149.jpg'>

<img src='content_image/304147.jpg'>

