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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 86 open journal systems 


Limiting spectral distribution of circulant type matrices with dependent inputs

Arup Bose, Indian Statistical Institute, Kolkata
Rajat Subhra Hazra, Indian Statistical Institute, Kolkata
Koushik Saha, Indian Statistical Institute, Kolkata


Abstract
Limiting spectral distribution (LSD) of scaled eigenvalues of circulant, symmetric circulant and a class of k-circulant matrices are known when the input sequence is independent and identically distributed with finite moments of suitable order. We derive the LSD of these matrices when the input sequence is a stationary, two sided moving average process of infinite order. The limits are suitable mixtures of normal, symmetric square root of the chisquare, and other mixture distributions, with the spectral density of the process involved in the mixtures.


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Pages: 2463-2491

Published on: November 9, 2009
Bibliography
  1. Bai, Z. D. Methodologies in spectral analysis of large-dimensional random matrices, a review. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. Statist. Sinica 9 (1999), no. 3, 611--677. MR1711663 (2000e:60044).

  2. Bai, Zhidong; Zhou, Wang. Large sample covariance matrices without independence structures in columns. Statist. Sinica 18 (2008), no. 2, 425--442. MR2411613 (2009d:60075).

  3. Bhatia, Rajendra. Matrix analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997. xii+347 pp. ISBN: 0-387-94846-5 MR1477662 (98i:15003).

  4. Bhattacharya, R. N.; Ranga Rao, R. Normal approximation and asymptotic expansions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, 1976. xiv+274 pp. MR0436272 (55 #9219).

  5. Bose, Arup; Mitra, Joydip. Limiting spectral distribution of a special circulant. Statist. Probab. Lett. 60 (2002), no. 1, 111--120. MR1945684 (2003j:60043).

  6. Bose, Arup; Mitra, Joydip; Sen, Arnab. Large dimensional random k-circulants. Technical Report No.R10/2008, Stat-Math Unit, Indian Statistical Institute, Kolkata. Submitted for publication.

  7. Bose, Arup; Sen, Arnab. Another look at the moment method for large dimensional random matrices. Electron. J. Probab. 13 (2008), no. 21, 588--628. MR2399292 (2009d:60049).

  8. Brockwell, Peter J.; Davis, Richard A. Introduction to time series and forecasting. Second edition. With 1 CD-ROM (Windows). Springer Texts in Statistics. Springer-Verlag, New York, 2002. xiv+434 pp. ISBN: 0-387-95351-5 MR1894099 (2002m:62002).

  9. Bryc, W?odzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1--38. MR2206341 (2007c:60039).

  10. Chatterjee, Sourav. A generalization of the Lindeberg principle. Ann. Probab. 34 (2006), no. 6, 2061--2076. MR2294976 (2008c:60028).

  11. Fan, Jianqing; Yao, Qiwei. Nonlinear time series. Nonparametric and parametric methods. Springer Series in Statistics. Springer-Verlag, New York, 2003. xx+551 pp. ISBN: 0-387-95170-9 MR1964455 (2004a:62002).

  12. Meckes, Mark W. Some results on random circulant matrice. arXiv:0902.2472v1 [math.PR], 2009.

  13. Massey, Adam; Miller, Steven J.; Sinsheimer, John. Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices. J. Theoret. Probab. 20 (2007), no. 3, 637--662. MR2337145 (2008f:15082).

  14. Zhou, Jin Tu. A formula solution for the eigenvalues of $g$-circulant matrices. (Chinese) Math. Appl. (Wuhan) 9 (1996), no. 1, 53--57. MR1384201 (97d:15017).













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Electronic Journal of Probability. ISSN: 1083-6489