Convergence of the spectrum of empirical covariance matrices for independent MRW processes
Allez, Romain1, 2 ; Rhodes, Rémi1, 2 ; Vargas, Vincent1, 2
1 CNRS, UMR 7534, 75016 Paris, France.
2 Université Paris-Dauphine, Ceremade, 75016 Paris, France.
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 327-360.

We study the asymptotic of the spectral distribution for large empirical covariance matrices composed of independent lognormal Multifractal Random Walk processes. The asymptotic is taken as the observation lag shrinks to 0. In this setting, we show that there exists a limiting spectral distribution whose Stieltjes transform is uniquely characterized by equations which we specify. We also illustrate our results by numerical simulations.

Reçu le :
DOI : 10.1051/ps/2014028
Classification : 60B20, 60G18, 60G15, 91G99
Mots clés : Multifractals, Marchenko-Pastur theorem, random matrices, Gaussian multiplicative chaos
Allez, Romain 1, 2 ; Rhodes, Rémi 1, 2 ; Vargas, Vincent 1, 2

1 CNRS, UMR 7534, 75016 Paris, France.
2 Université Paris-Dauphine, Ceremade, 75016 Paris, France. 
@article{PS_2015__19__327_0,
     author = {Allez, Romain and Rhodes, R\'emi and Vargas, Vincent},
     title = {Convergence of the spectrum of empirical covariance matrices for independent {MRW} processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {327--360},
     publisher = {EDP-Sciences},
     volume = {19},
     year = {2015},
     doi = {10.1051/ps/2014028},
     zbl = {1331.60015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2014028/}
}
TY  - JOUR
AU  - Allez, Romain
AU  - Rhodes, Rémi
AU  - Vargas, Vincent
TI  - Convergence of the spectrum of empirical covariance matrices for independent MRW processes
JO  - ESAIM: Probability and Statistics
PY  - 2015
SP  - 327
EP  - 360
VL  - 19
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2014028/
DO  - 10.1051/ps/2014028
LA  - en
ID  - PS_2015__19__327_0
ER  - 
%0 Journal Article
%A Allez, Romain
%A Rhodes, Rémi
%A Vargas, Vincent
%T Convergence of the spectrum of empirical covariance matrices for independent MRW processes
%J ESAIM: Probability and Statistics
%D 2015
%P 327-360
%V 19
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2014028/
%R 10.1051/ps/2014028
%G en
%F PS_2015__19__327_0
Allez, Romain; Rhodes, Rémi; Vargas, Vincent. Convergence of the spectrum of empirical covariance matrices for independent MRW processes. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 327-360. doi : 10.1051/ps/2014028. http://www.numdam.org/articles/10.1051/ps/2014028/

E. Bacry, A. Kozhemyak and J.F. Muzy, Log-normal continuous cascade model of asset returns: aggregation properties and estimation. Quant. Finance 13 (2013) 795–818. | Zbl

E. Bacry and J.F. Muzy, Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 (2003) 449–475. | Zbl

S. Belinschi, A. Dembo and A. Guionnet, Spectral measure of heavy tailed band and covariance random matrices. Comm. Math. Phys. 289 (2009) 1023–1055. | Zbl

G. Ben Arous and A. Guionnet, The spectrum of heavy-tailed random matrices. Comm. Math. Phys. 278 (2008) 715–751. | Zbl

J.P. Bouchaud, A. Matacz and M. Potters, Leverage effect in financial markets: The retarded volatility model. Phys. Rev. Lett. 87 (2001) 228701.

J.P. Bouchaud and M. Potters, Financial Applications of Random Matrix Theory: A Short Review. In Oxf. Handb. Random Matrix Theory. Oxford University Press (2011). | Zbl

J.P. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing. Cambridge University Press, Cambridge (2003). | Zbl

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues. Quant. Finance 1 (2001) 223–236. | Zbl

J.S. Geronimo, T.P. Hill, Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform. J. Approx. Theory 121 (2003) 54–60. | Zbl

A. Khorunzhy, B. Khoruzhenko, L. Pastur and M. Shcherbina, The large-n limit in statistical mechanics and the spectral theory of disordered systems. Phase Transitions and Critical Phenomena. In vol. 73. Academic Press, New-York (1992).

Y. Li and X. Zheng, On the estimation of integrated covariance matrices of high dimensional diffusion processes. Ann. Statist. 39 (2011) 3121–3151. | Zbl

V.A. Marchenko and L.A. Pastur, Distribution of eigenvalues for some sets of random matrices. Math. USSR-Sb. 1 (1967) 457–483. | Zbl

R. Rhodes and V. Vargas., Gaussian multiplicative chaos and applications: a review. ESAIM: PS 11 (2014) 315–392. | Zbl

C. Robert and M. Rosenbaum, On the limiting spectral distribution of the covariance matrices of time-lagged processes. J. Multivar. Anal. 101 (2010) 2434–2451. | Zbl

M. Potters, J.-P. Bouchaud and L. Laloux, Financial applications of random matrix theory: Old laces and new pieces. Acta Physica Polonica B 36 (2005) 2767. | Zbl

J. Duchon, R. Robert and V. Vargas, Forecasting volatility with the multifractal random walk model. Math. Finance 22 (2012) 83–108. | Zbl

Cité par Sources :