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. 
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     author = {Allez, Romain and Rhodes, R\'emi and Vargas, Vincent},
     title = {Convergence of the spectrum of empirical covariance matrices for independent {MRW} processes},
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     pages = {327--360},
     publisher = {EDP-Sciences},
     volume = {19},
     year = {2015},
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     zbl = {1331.60015},
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     url = {http://archive.numdam.org/articles/10.1051/ps/2014028/}
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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://archive.numdam.org/articles/10.1051/ps/2014028/

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