Product of exponentials and spectral radius of random k-circulants
Bose, Arup ; Hazra, Rajat Subhra ; Saha, Koushik
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 424-443 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/10-AIHP404
Classification:  Primary 60B20,  secondary,  60B10,  60F05,  62E20,  62G32,  15A52,  60F99,  60F05 
@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},
     volume = {48},
     year = {2012},
     pages = {424-443},
     doi = {10.1214/10-AIHP404},
     zbl = {1244.60010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 424-443. doi : 10.1214/10-AIHP404. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_2_424_0/

[1] R. Adamczak. A few remarks on the operator norm of random Toeplitz matrices. J. Theoret. Probab. 23 (2008) 85-108. | MR 2591905 | Zbl 1191.15030

[2] A. Auffinger, G. Ben Arous and S. Peche. Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 859-610. | Numdam | MR 2548495 | Zbl 1177.15037

[3] Z. D. Bai, J. Silverstein and Y. Q. Yin. A note on the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivariate Anal. 26 (1988) 166-168. | MR 963829 | Zbl 0652.60040

[4] Z. D. Bai and Y. Q. Yin. 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 863545 | Zbl 0586.60021

[5] Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16 (1988) 1729-1741. | MR 958213 | Zbl 0677.60038

[6] Z. D. Bai and Y. Q. Yin. Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21 (1993) 1275-1294. | MR 1235416 | Zbl 0779.60026

[7] A. Bose, R. S. Hazra and K. Saha. Spectral norm of circulant type matrices. J. Theoret. Probab. 24 (2011) 479-516. | MR 2795050 | Zbl 1241.60006

[8] A. Bose, R. S. Hazra and K. Saha. Spectral norm of circulant type matrices with heavy tailed entries. Electron. Comm. Probab. 15 (2010) 299-313. | MR 2670197 | Zbl 1226.60008

[9] A. Bose, J. Mitra and A. Sen. Large dimensional random k circulants. Technical Report R10/2008, Stat-Math Unit, Indian Statistical Institute, Kolkata, 2008. Available at www.isical.ac.in/~statmath.

[10] A. Bose, J. Mitra and A. Sen. Large dimensional random k circulants. J. Theoret. Probab. (2012). To appear. DOI:10.1007/s10959-010-0312-9. | MR 2956212

[11] A. Bose and A. Sen. Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices. Electron. Comm. Probab. 12 (2007) 29-35. | MR 2284045 | Zbl 1130.60042

[12] W. Bryc, A. Dembo and T. Jiang. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006) 1-38. | MR 2206341 | Zbl 1094.15009

[13] W. Bryc and S. Sethuraman. 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 2797947 | Zbl 1243.60006

[14] J. Chen and M.-Q. Liu. Optimal mixed-level k-circulant supersaturated designs. J. Statist. Plann. Inference 138 (2008) 4151-4157. | MR 2455995 | Zbl 1146.62055

[15] R. A. Davis and T. Mikosch. The maximum of the periodogram of a non-Gaussian sequence. Ann. Probab. 27 (1999) 522-536. | MR 1681157 | Zbl 1073.62556

[16] P. Embrechts, C. Kluppelberg and T. Mikosch. Modelling Extremal Events in Insurance and Finance. Springer, Berlin, 1997. | MR 1458613 | Zbl 0873.62116

[17] A. Erdélyi. Asymptotics Expansions. Dover, New York, 1956. | MR 78494 | Zbl 0070.29002

[18] J. Galambos and I. Simonelli. Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions. Marcel Dekker, New York, 2004. | MR 2077249 | Zbl 1172.60300

[19] S. Geman. A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980) 252-261. | MR 566592 | Zbl 0428.60039

[20] S. Geman. The spectral radius of large random matrices. Ann. Probab. 14 (1986) 1318-1328. | MR 866352 | Zbl 0605.60037

[21] S. Georgiou and C. Koukouvinos. Multi-level k-circulant supersaturated designs. Metrika 64 (2006) 209-220. | MR 2259223 | Zbl 1100.62076

[22] E. Kostlan. On the spectra of Gaussian matrices. Linear Algebra Appl. 162/164 (1992) 385-388. | MR 1148410 | Zbl 0748.15024

[23] Z. Lin and W. Liu. On maxima of periodograms of stationary processes. Ann. Statist. 37 (2009) 2676-2695. | MR 2541443 | Zbl 1206.62017

[24] Z. A. Lomnicki. On the distribution of products of random variables. J. Roy. Statist. Soc. Ser. B 29 (1967) 513-524. | MR 222928 | Zbl 0161.37801

[25] M. W. Meckes. On the spectral norm of a random Toeplitz matrix. Electron. Comm. Probab. 12 (2007) 315-325. | MR 2342710 | Zbl 1130.15018

[26] S. Resnick. Extreme Values, Regular Variation and Point Processes. Applied Probability: A Series of the Applied Probability Trust 4. Springer, New York, 1987. | MR 900810 | Zbl 0633.60001

[27] B. Rider. A limit theorem at the edge of a non-Hermitian random matrix ensemble. J. Phys. A 36 (2003) 3401-3409. | MR 1986426 | Zbl 1039.60037

[28] H. Rootzèn. Extreme value theory for moving average processes. Ann. Probab. 14 (1986) 612-652. | MR 832027 | Zbl 0604.60019

[29] J. W. Silverstein. The smallest eigenvalue of a large dimensional Wishart matrix. Ann. Probab. 13 (1985) 1364-1368. | MR 806232 | Zbl 0591.60025

[30] J. W. Silverstein. On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivariate Anal. 30 (1989) 307-311. | MR 1015375 | Zbl 0692.60022

[31] J. W. Silverstein. The spectral radii and norms of large-dimensional non-central random matrices. Comm. Statist. Stochastic Models 10 (1994) 525-532. | MR 1284550 | Zbl 0806.15018

[32] A. Soshnikov. Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 (2004) 82-91. | MR 2081462 | Zbl 1060.60013

[33] A. Soshnikov. 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 2234922 | Zbl 1169.15302

[34] M. D. Springer and W. E. Thompson. The distribution of products of Beta, Gamma and Gaussian random variables. SIAM J. Appl. Math. 18 (1970) 721-737. | MR 264733 | Zbl 0198.23703

[35] V. V. Strok. Circulant matrices and the spectra of de Bruijn graphs. Ukrainian Math. J. 44 (1992) 1446-1454. | MR 1213901 | Zbl 0788.05068

[36] Q. Tang. From light tails to heavy tails through multiplier. Extremes 11 (2008) 379-391. | MR 2453346 | Zbl 1199.60040

[37] C. A. Tracy and H. Widom. 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 1844228

[38] A. M. Walker. Some asymtotic results for the periodogram of a stationary time series. J. Austral. Math. Soc. 5 (1965) 107-128. | MR 177457 | Zbl 0128.38701

[39] Y. K. Wu, R. Z. Jia and Q. Li. g-circulant solutions to the (0, 1) matrix equation Am = Jn. Linear Algebra Appl. 345 (2002) 195-224. | MR 1883273 | Zbl 0998.15015

[40] Y. Q. Yin, Z. D. Bai and P. R. Krishnaiah. Limiting behavior of the eigenvalues of a multivariate F matrix. J. Multivariate Anal. 13 (1983) 508-516. | MR 727036 | Zbl 0531.62018

[41] Y. Q. Yin, Z. D. Bai and P. R. Krishnaiah. On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 (1988) 509-521. | MR 950344 | Zbl 0627.62022

[42] J. T. Zhou. A formula solution for the eigenvalues of g circulant matrices. Math. Appl. (Wuhan) 9 (1996) 53-57. | MR 1384201 | Zbl 0947.15500