## Electronic Communications in Probability

### Deviation inequalities and moderate deviations for estimators of parameters in an Ornstein-Uhlenbeck process with linear drift

#### Abstract

Some deviation inequalities and moderate deviation principles for the maximum likelihood estimators of parameters in an Ornstein-Uhlenbeck process with linear drift are established by the logarithmic Sobolev inequality and the exponential martingale method.

#### Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 21, 210-223.

Dates
Accepted: 24 May 2009
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465234730

Digital Object Identifier
doi:10.1214/ECP.v14-1466

Mathematical Reviews number (MathSciNet)
MR2507750

Zentralblatt MATH identifier
1189.60058

Subjects
Primary: 60F12
Secondary: 62F12: Asymptotic properties of estimators 62N02: Estimation

Rights

#### Citation

Gao, Fuqing; Jiang, Hui. Deviation inequalities and moderate deviations for estimators of parameters in an Ornstein-Uhlenbeck process with linear drift. Electron. Commun. Probab. 14 (2009), paper no. 21, 210--223. doi:10.1214/ECP.v14-1466. https://projecteuclid.org/euclid.ecp/1465234730

#### References

• Bercu, B.; Rouault, A. Sharp large deviations for the Ornstein-Uhlenbeck process. Teor. Veroyatnost. i Primenen. 46 (2001), no. 1, 74–93; translation in Theory Probab. Appl. 46 (2002), no. 1, 1–19
• Bobkov, S. G.; Götze, F. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999), no. 1, 1–28.
• Cattiaux, Patrick; Guillin, Arnaud. Deviation bounds for additive functionals of Markov processes. ESAIM Probab. Stat. 12 (2008), 12–29 (electronic).
• Dembo, A. Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab. 1 (1996), no. 3, 11–17 (electronic).
• Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
• Deuschel, Jean-Dominique; Stroock, Daniel W. Large deviations.Pure and Applied Mathematics, 137. Academic Press, Inc., Boston, MA, 1989. xiv+307 pp. ISBN: 0-12-213150-9
• Djellout, H.; Guillin, A.; Wu, L. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (2004), no. 3B, 2702–2732.
• Djellout, H.; Guillin, A.; Wu, L. Moderate deviations of empirical periodogram and non-linear functionals of moving average processes. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 4, 393–416.
• Florens-Landais, Danielle; Pham, Huyên. Large deviations in estimation of an Ornstein-Uhlenbeck model. J. Appl. Probab. 36 (1999), no. 1, 60–77.
• Gourcy, Mathieu; Wu, Liming. Logarithmic Sobolev inequalities of diffusions for the $L\sp 2$ metric. Potential Anal. 25 (2006), no. 1, 77–102.
• Guillin, A.; Liptser, R. Examples of moderate deviation principle for diffusion processes. Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 4, 803–828 (electronic).
• Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120–216, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
• Ledoux, Michel. The concentration of measure phenomenon.Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9
• Lezaud, Pascal. Chernoff and Berry-Esséen inequalities for Markov processes. ESAIM Probab. Statist. 5 (2001), 183–201 (electronic).
• Kutoyants, Yury A. Statistical inference for ergodic diffusion processes.Springer Series in Statistics. Springer-Verlag London, Ltd., London, 2004. xiv+481 pp. ISBN: 1-85233-759-1
• Prakasa Rao, B. L. S. Statistical Inference for Diffusion Type Processes. Oxford University Presss, New York,1999.
• Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1991. x+533 pp. ISBN: 3-540-52167-4
• Wu, Liming. A deviation inequality for non-reversible Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 435–445.