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104
A. C. ROSA AND M. E. NOGUEIRA
Hence, Azuma’s inequality (cf. Delecroix et al. [11]) leads to
Xn ✏ ! P U(i) (x,y) >
( ✏ 2 T n 2 h 2 d )  2exp Tn
aswellas
P
=2exp Tn , 32 k⇤2
Xn
`=2
✏! (x, y) >
(
( ✏2Tnh2d)
)
⇤

V (i) ` 1,Tn
✏2 Tn2 h2d Tn
k,Tn 2 k=1
32k⇤2n 2
( ✏2Tnh2d)

2exp Tn .
2 exp

32 k⇤2 (n 1) 2 32 k⇤2
Then, taking C1 = 1 32 k⇤2
2
, we have the announced result.
5. Final remarks
A straightforward improvement of Theorem 3.2 is the almost sure consis- tency of q↵,T(x), uniformly on x belonging to a compact subset of Rd. In line with the works of Didi and Louani [15] and Didi [13], we would deduce the almost sure convergence of the corresponding nonparametric predictor. In addition, it will be interesting to study consistency properties with rates of convergence.
Acknowledgments
We kindly acknowledge the support of the Department of Mathematics of the University of Coimbra. We are also grateful to the referee for his valuable suggestions and comments which greatly improved the text.
References
[1] G. Banon, Nonparametric identification for di↵usion processes, SIAM J. Control Optim. 16 (3), 380–395, 1978.
[2] V. Bergelson, A. Leibman, and C. G. Moreira, From discrete to continuous-time ergodic theorems, Ergodic Theory Dynam. Systems 32 (2), 383–426, 2012.
[3] D. Blanke and D. Bosq, Regression estimation and prediction in continuous time, J. Japan Statist. Soc. 38 (1), 15–26, 2008.
[4] D. Bosq, Nonparametric statistics for stochastic processes, Springer-Verlag, New York, 1998.
[5] D. Bosq and D. Blanke, Inference and prediction in large dimensions, John Wiley and Sons, Chichester, 2007.