Moderate deviations and central limit theorem for small perturbation Wishart processes

Abstract

Let X^ϵ be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process X^ϵ is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: \mathrm{d}{X}_t^{ϵ}=\sqrt{ϵX_t^{ϵ}}\mathrm{d}{B}_t+\mathrm{d}{B}_{\mathit{t}}^{\mathit{\text{'}}}\sqrt{ϵX_t^{ϵ}+\rho I_m\mathrm{d}t}, X₀ = x, where B is an m × m matrix valued Brownian motion and B′denotes the transpose of the matrix B. In this paper, we prove that \{X_t^{ϵ}-X_t^{\mathrm{0}}/\sqrt{ϵh^{\mathrm{2}}(ϵ)},ϵ>\mathrm{0}\} satisfies a large deviation principle, and (X_t^{ϵ}-X_t^{\mathrm{0}})/\sqrt{ϵ} converges to a Gaussian process, where h(ϵ)\to +\infty and \sqrt{ϵ}h(ϵ)\to \mathrm{0} as ϵ\to \mathrm{0}. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of X^ϵ are also obtained by the delta method.

Key words： Large deviation; moderate deviation; central limit theorem; Wishart process; eigenvalue

References

1 Bru M F. Diffusions of perturbed principal component analysis. J Multivariate Anal, 1989, 29(1): 127-136

2 Bru M F. Wishart processes. J Theoret Probab, 1991, 4(4): 725-751

3 Dembo A, Zeitouni O. Large Deviations Techniques and Applications. Berlin: Springer-Verlag, 2009

4 Donati-Martin C. Large deviations for Wishart processes. Probab Math Statist, 2008, 28(2): 325-343

5 Donati-Martin C, Doumerc Y, Matsumoto H, Yor M. Some properties of the Wishart processes and a matrix extension of the Hartman-Watson laws. Universités de Paris 6 and Paris 7-CNRS (UMR 7599), 2003

6 Donati-Martin C, Rouault A, Yor M, Zani M. Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes. Probab Theory Related Fields, 2004, 129(2): 261-289

7 Gao F Q, Zhao X Q. Delta method in large deviations and moderate deviations for estimators. Ann Statist, 2011, 39(2): 1211-1240

8 Guillin A. Averaging principle of SDE with small diffusion: Moderate deviations. Ann Probab, 2003, 31(1): 413-443

9 Guionnet A. Large Random Matrices: Lectures on Mcroscopic Asymptotics. Berlin: Springer, 2009

10 Ma Y T, Wang R, Wu L M. Moderate deviation principle for dynamical systems with small random perturbation. arXiv: 1107.3432, July 2011

11 Muirhead R J. Aspects of Multivariate Statistical Theory. New York: Wiley, 1982

12 Revuz D, Yor M. Continuous Martingales and Brownian Motion. Berlin: Springer-Verlag, 1999

13 Stewart G W, Sun J. Matrix Perturbation Theory. New York: Academic Press, 1990

14 Van der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes with Application to Statistics. New York: Springer-Verlag, 1996