Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Product of exponentials and spectral radius of random k-circulants

Arup Bose, Rajat Subhra Hazra, and Koushik Saha

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 424-443.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148206

Digital Object Identifier
doi:10.1214/10-AIHP404

Mathematical Reviews number (MathSciNet)
MR2954262

Zentralblatt MATH identifier
1244.60010

Subjects
Primary: Primary 60B20 secondary 60B10 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory 62G32: Statistics of extreme values; tail inference 15A52 60F99: None of the above, but in this section 60F05: Central limit and other weak theorems

Keywords
Eigenvalues Gumbel distribution k-circulant matrix Laplace asymptotics Large dimensional random matrix Linear process Normal approximation Spectral radius Spectral density Tail of product

Citation

Bose, Arup; Hazra, Rajat Subhra; Saha, Koushik. Product of exponentials and spectral radius of random k -circulants. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 424--443. doi:10.1214/10-AIHP404. https://projecteuclid.org/euclid.aihp/1334148206


Export citation

References

  • [1] R. Adamczak. A few remarks on the operator norm of random Toeplitz matrices. J. Theoret. Probab. 23 (2008) 85–108.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [7] A. Bose, R. S. Hazra and K. Saha. Spectral norm of circulant type matrices. J. Theoret. Probab. 24 (2011) 479–516.
  • [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.
  • [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.
  • [11] A. Bose and A. Sen. Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices. Electron. Comm. Probab. 12 (2007) 29–35.
  • [12] W. Bryc, A. Dembo and T. Jiang. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006) 1–38.
  • [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.
  • [14] J. Chen and M.-Q. Liu. Optimal mixed-level k-circulant supersaturated designs. J. Statist. Plann. Inference 138 (2008) 4151–4157.
  • [15] R. A. Davis and T. Mikosch. The maximum of the periodogram of a non-Gaussian sequence. Ann. Probab. 27 (1999) 522–536.
  • [16] P. Embrechts, C. Kluppelberg and T. Mikosch. Modelling Extremal Events in Insurance and Finance. Springer, Berlin, 1997.
  • [17] A. Erdélyi. Asymptotics Expansions. Dover, New York, 1956.
  • [18] J. Galambos and I. Simonelli. Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions. Marcel Dekker, New York, 2004.
  • [19] S. Geman. A limit theorem for the norm of random matrices. Ann. Probab. 8 (1980) 252–261.
  • [20] S. Geman. The spectral radius of large random matrices. Ann. Probab. 14 (1986) 1318–1328.
  • [21] S. Georgiou and C. Koukouvinos. Multi-level k-circulant supersaturated designs. Metrika 64 (2006) 209–220.
  • [22] E. Kostlan. On the spectra of Gaussian matrices. Linear Algebra Appl. 162/164 (1992) 385–388.
  • [23] Z. Lin and W. Liu. On maxima of periodograms of stationary processes. Ann. Statist. 37 (2009) 2676–2695.
  • [24] Z. A. Lomnicki. On the distribution of products of random variables. J. Roy. Statist. Soc. Ser. B 29 (1967) 513–524.
  • [25] M. W. Meckes. On the spectral norm of a random Toeplitz matrix. Electron. Comm. Probab. 12 (2007) 315–325.
  • [26] S. Resnick. Extreme Values, Regular Variation and Point Processes. Applied Probability: A Series of the Applied Probability Trust 4. Springer, New York, 1987.
  • [27] B. Rider. A limit theorem at the edge of a non-Hermitian random matrix ensemble. J. Phys. A 36 (2003) 3401–3409.
  • [28] H. Rootzèn. Extreme value theory for moving average processes. Ann. Probab. 14 (1986) 612–652.
  • [29] J. W. Silverstein. The smallest eigenvalue of a large dimensional Wishart matrix. Ann. Probab. 13 (1985) 1364–1368.
  • [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.
  • [31] J. W. Silverstein. The spectral radii and norms of large-dimensional non-central random matrices. Comm. Statist. Stochastic Models 10 (1994) 525–532.
  • [32] A. Soshnikov. Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Comm. Probab. 9 (2004) 82–91.
  • [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.
  • [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.
  • [35] V. V. Strok. Circulant matrices and the spectra of de Bruijn graphs. Ukrainian Math. J. 44 (1992) 1446–1454.
  • [36] Q. Tang. From light tails to heavy tails through multiplier. Extremes 11 (2008) 379–391.
  • [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.
  • [38] A. M. Walker. Some asymtotic results for the periodogram of a stationary time series. J. Austral. Math. Soc. 5 (1965) 107–128.
  • [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.
  • [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.
  • [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.
  • [42] J. T. Zhou. A formula solution for the eigenvalues of g circulant matrices. Math. Appl. (Wuhan) 9 (1996) 53–57.