Asymptotics for the ${L}^{p}$-deviation of the variance estimator under diffusion
ESAIM: Probability and Statistics, Volume 8  (2004), p. 132-149

We consider a diffusion process ${X}_{t}$ smoothed with (small) sampling parameter $\epsilon$. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\stackrel{^}{\alpha }}_{\epsilon }$ with window $h\left(\epsilon \right)$ of a function $\alpha$ of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the ${L}^{p}$ deviations such as $\phantom{\rule{-28.45274pt}{0ex}}\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\stackrel{^}{\alpha }}_{\epsilon }-\alpha ∥}_{p}^{p}-𝔼{∥{\stackrel{^}{\alpha }}_{\epsilon }-\alpha ∥}_{p}^{p}\right).$

DOI : https://doi.org/10.1051/ps:2004005
Classification:  60F05,  60F25,  60J60,  60H05,  62M02,  62M05
Keywords: variance estimator, kernel, ${L}^{p}$-deviation, central limit theorem
@article{PS_2004__8__132_0,
author = {Doukhan, Paul and Le\'on, Jos\'e R.},
title = {Asymptotics for the $L^p$-deviation of the variance estimator under diffusion},
journal = {ESAIM: Probability and Statistics},
publisher = {EDP-Sciences},
volume = {8},
year = {2004},
pages = {132-149},
doi = {10.1051/ps:2004005},
zbl = {pre02161879},
mrnumber = {2085611},
language = {en},
url = {http://www.numdam.org/item/PS_2004__8__132_0}
}

Doukhan, Paul; León, José R. Asymptotics for the $L^p$-deviation of the variance estimator under diffusion. ESAIM: Probability and Statistics, Volume 8 (2004) , pp. 132-149. doi : 10.1051/ps:2004005. http://www.numdam.org/item/PS_2004__8__132_0/

[1] J. Beirlant and D.M. Mason, On the asymptotic normality of the ${L}^{p}$-norm of empirical functional. Math. Methods Statist. 4 (1995) 1-19. | Zbl 0831.62019

[2] C. Berzin-Joseph, J.R. León and J. Ortega, Non-linear functionals of the Brownian bridge and some applications. Stoch. Proc. Appl. 92 (2001) 11-30. | Zbl 1047.60082

[3] P. Brugière, Théorème de limite centrale pour un estimateur non paramétrique de la variance d'un processus de diffusion multidimensionnelle. Ann. Inst. Henri Poincaré, Probab. Stat. 29 (1993) 357-389. | Numdam | Zbl 0792.60017

[4] P.D. Ditlevsen, S. Ditlevsen and K.K. Andersen, The fast climate fluctuations during the stadial and interstadial climate states. Ann. Glaciology 35 (2002).

[5] P. Doukhan, J.R. León and F. Portal, Calcul de la vitesse de convergence dans le théorème central limite vis-à-vis des distances de Prohorov, Dudley et Lévy dans le cas de v. a. dépendantes. Probab. Math. Statist. 6 (1985) 19-27. | Zbl 0607.60019

[6] V. Genon-Catalot, C. Laredo and D. Picard, Non-parametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Statist. 19 (1992) 317-335. | Zbl 0776.62033

[7] I.J. Gihman and A.V. Skorohov, Stochastic differential equations. Springer-Verlag, Berlin, New York (1972). | MR 346904 | Zbl 0242.60003

[8] E. Giné, D. Mason and Yu. Zaitsev, The ${L}^{1}$-norm density estimator process. To appear in Ann. Prob. | Zbl 1031.62026

[9] A. Gloter, Parameter estimation for a discrete sampling of an integrated Ornstein-Uhlenbeck process. Statistics 35 (2000) 225-243. | Zbl 0980.62072

[10] J. Jacod, On continuous conditional martingales and stable convergence in law, sémin. Probab. XXXI, LNM 1655, Springer (1997) 232-246. | Numdam | Zbl 0884.60038

[11] P. Major, Multiple Wiener-Itô integrals. Springer-Verlag, New York, Lect. Notes Math. 849 (1981). | MR 611334 | Zbl 0451.60002

[12] G. Perera and M. Wschebor, Crossings and occupation measures for a class of semimartingales. Ann. Probab. 26 (1998) 253-266. | Zbl 0943.60019

[13] G. Perera and M. Wschebor, Inference on the variance and smoothing of the paths of diffusions. Ann. Inst. Henri Poincaré, Probab. Stat. 38 (2002) 1009-1022. | Numdam | Zbl 1011.62083

[14] E. Rio, About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1995) 35-61. | Numdam | Zbl 0869.60021

[15] H.P. Rosenthal, On the subspaces of ${L}^{p}$, ($p>2$) spanned by sequences of independent random variables. Israël Jour. Math. 8 (1970) 273-303. | Zbl 0213.19303

[16] V.V. Shergin, On the convergence rate in the central limit theorem for $m$-dependent random variables. Theor. Proba. Appl. 24 (1979) 782-796. | Zbl 0447.60023

[17] P. Soulier, Non-parametric estimation of the diffusion coefficient of a diffusion process. Stoch. Anal. Appl. 16 (1998) 185-200. | Zbl 0894.62093

[18] G. Terdik, Bilinear Stochastic Models and Related problems of Nonlinear Time Series. Springer-Verlag, New York, Lect. Notes Statist. 142 (1999). | MR 1702281 | Zbl 0928.62068