Convergence of functional k-nearest neighbor regression estimate with functional responses

Let (X ¹, Y ¹),..., (X ⁿ, Y ⁿ) be independent and identically dis-tributed random elements taking values in F×H, where F is a semi-metric space and H is a separable Hilbert space. We investigate the rates of strong (almost sure) convergence of the k-nearest neighbor estimate. We give two convergence results assuming a finite moment condition and exponential tail condition on the noises respectively, with the latter requiring less stringent conditions on k for convergence.