Nous considérons des matrices aléatoires k-circulantes de taille n × n avec n → ∞ et k = k(n), dont les entrées {al}l≥0 sont des variables aléatoires, de moment (2 + δ) fini, indépendantes et identiquement distribuées. Nous étudions la distribution asymptotique du rayon spectral, lorsque n = kg + 1. Pour établir cette distribution asymptotique, nous calculons d'abord le comportement de la queue du produit de g variables aléatoires exponentielles i.i.d. Ensuite, en utilisant un résultat sur le comportement des queues et les techniques appropriées d'approximation normale, nous montrons que, après renormalisation et recentrage, la distribution limite est une distribution de Gumbel. Nous identifions explicitement les constantes de recentrage et de remise à l'échelle.
We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {al}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.
Mots clés : eigenvalues, Gumbel distribution, k-circulant matrix, Laplace asymptotics, large dimensional random matrix, linear process, normal approximation, spectral radius, spectral density, tail of product
@article{AIHPB_2012__48_2_424_0, author = {Bose, Arup and Hazra, Rajat Subhra and Saha, Koushik}, title = {Product of exponentials and spectral radius of random $k$-circulants}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {424--443}, publisher = {Gauthier-Villars}, volume = {48}, number = {2}, year = {2012}, doi = {10.1214/10-AIHP404}, zbl = {1244.60010}, language = {en}, url = {http://www.numdam.org/articles/10.1214/10-AIHP404/} }
TY - JOUR AU - Bose, Arup AU - Hazra, Rajat Subhra AU - Saha, Koushik TI - Product of exponentials and spectral radius of random $k$-circulants JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 424 EP - 443 VL - 48 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP404/ DO - 10.1214/10-AIHP404 LA - en ID - AIHPB_2012__48_2_424_0 ER -
%0 Journal Article %A Bose, Arup %A Hazra, Rajat Subhra %A Saha, Koushik %T Product of exponentials and spectral radius of random $k$-circulants %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 424-443 %V 48 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP404/ %R 10.1214/10-AIHP404 %G en %F AIHPB_2012__48_2_424_0
Bose, Arup; Hazra, Rajat Subhra; Saha, Koushik. Product of exponentials and spectral radius of random $k$-circulants. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 424-443. doi : 10.1214/10-AIHP404. http://www.numdam.org/articles/10.1214/10-AIHP404/
[1] A few remarks on the operator norm of random Toeplitz matrices. J. Theoret. Probab. 23 (2008) 85-108. | MR | Zbl
.[2] Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 859-610. | Numdam | MR | Zbl
, and .[3] A note on the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivariate Anal. 26 (1988) 166-168. | MR | Zbl
, and .[4] Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang. Probab. Theory Related Fields 73 (1986) 555-569. | MR | Zbl
and .[5] Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16 (1988) 1729-1741. | MR | Zbl
and .[6] Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21 (1993) 1275-1294. | MR | Zbl
and .[7] Spectral norm of circulant type matrices. J. Theoret. Probab. 24 (2011) 479-516. | MR | Zbl
, and .[8] Spectral norm of circulant type matrices with heavy tailed entries. Electron. Comm. Probab. 15 (2010) 299-313. | MR | Zbl
, and .[9] Large dimensional random k circulants. Technical Report R10/2008, Stat-Math Unit, Indian Statistical Institute, Kolkata, 2008. Available at www.isical.ac.in/~statmath.
, and .[10] Large dimensional random k circulants. J. Theoret. Probab. (2012). To appear. DOI:10.1007/s10959-010-0312-9. | MR
, and .[11] Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices. Electron. Comm. Probab. 12 (2007) 29-35. | MR | Zbl
and .[12] Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006) 1-38. | MR | Zbl
, and .[13] A remark on the maximum eigenvalue for circulant matrices. In High Dimensional Probabilities V: The Luminy Volume 179-184. IMS Collections 5. Institute of Mathematical Statistics, Beachwood, OH, 2009. | MR | Zbl
and .[14] Optimal mixed-level k-circulant supersaturated designs. J. Statist. Plann. Inference 138 (2008) 4151-4157. | MR | Zbl
and .[15] The maximum of the periodogram of a non-Gaussian sequence. Ann. Probab. 27 (1999) 522-536. | MR | Zbl
and .[16] Modelling Extremal Events in Insurance and Finance. Springer, Berlin, 1997. | MR | Zbl
, and .[17] Asymptotics Expansions. Dover, New York, 1956. | MR | Zbl
.[18] Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions. Marcel Dekker, New York, 2004. | MR | Zbl
and .[19] A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980) 252-261. | MR | Zbl
.[20] The spectral radius of large random matrices. Ann. Probab. 14 (1986) 1318-1328. | MR | Zbl
.[21] Multi-level k-circulant supersaturated designs. Metrika 64 (2006) 209-220. | MR | Zbl
and .[22] On the spectra of Gaussian matrices. Linear Algebra Appl. 162/164 (1992) 385-388. | MR | Zbl
.[23] On maxima of periodograms of stationary processes. Ann. Statist. 37 (2009) 2676-2695. | MR | Zbl
and .[24] On the distribution of products of random variables. J. Roy. Statist. Soc. Ser. B 29 (1967) 513-524. | MR | Zbl
.[25] On the spectral norm of a random Toeplitz matrix. Electron. Comm. Probab. 12 (2007) 315-325. | MR | Zbl
.[26] Extreme Values, Regular Variation and Point Processes. Applied Probability: A Series of the Applied Probability Trust 4. Springer, New York, 1987. | MR | Zbl
.[27] A limit theorem at the edge of a non-Hermitian random matrix ensemble. J. Phys. A 36 (2003) 3401-3409. | MR | Zbl
.[28] Extreme value theory for moving average processes. Ann. Probab. 14 (1986) 612-652. | MR | Zbl
.[29] The smallest eigenvalue of a large dimensional Wishart matrix. Ann. Probab. 13 (1985) 1364-1368. | MR | Zbl
.[30] On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivariate Anal. 30 (1989) 307-311. | MR | Zbl
.[31] The spectral radii and norms of large-dimensional non-central random matrices. Comm. Statist. Stochastic Models 10 (1994) 525-532. | MR | Zbl
.[32] Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 (2004) 82-91. | MR | Zbl
.[33] Poisson statistics for the largest eigenvalues in random matrix ensembles. In Mathematical Physics of Quantum Mechanics 351-364. Lecture Notes in Phys. 690. Springer, Berlin, 2006. | MR | Zbl
.[34] The distribution of products of Beta, Gamma and Gaussian random variables. SIAM J. Appl. Math. 18 (1970) 721-737. | MR | Zbl
and .[35] Circulant matrices and the spectra of de Bruijn graphs. Ukrainian Math. J. 44 (1992) 1446-1454. | MR | Zbl
.[36] From light tails to heavy tails through multiplier. Extremes 11 (2008) 379-391. | MR | Zbl
.[37] The distribution of the largest eigenvalue in the Gaussian ensembles: β = 1, 2, 4. In Calogero-Moser-Sutherland Models 461-472. Springer, New York, 2000. | MR
and .[38] Some asymtotic results for the periodogram of a stationary time series. J. Austral. Math. Soc. 5 (1965) 107-128. | MR | Zbl
.[39] g-circulant solutions to the (0, 1) matrix equation Am = Jn. Linear Algebra Appl. 345 (2002) 195-224. | MR | Zbl
, and .[40] Limiting behavior of the eigenvalues of a multivariate F matrix. J. Multivariate Anal. 13 (1983) 508-516. | MR | Zbl
, and .[41] On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 (1988) 509-521. | MR | Zbl
, and .[42] A formula solution for the eigenvalues of g circulant matrices. Math. Appl. (Wuhan) 9 (1996) 53-57. | MR | Zbl
.Cité par Sources :