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Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

Part of: Limit theorems Multivariate analysis Communication, information

Published online by Cambridge University Press:  01 July 2016

Anne Fey ,
Remco van der Hofstad  and
Marten J. Klok
Anne Fey*
Affiliation:
Delft University of Technology
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Marten J. Klok*
Affiliation:
Delft University of Technology
*
∗ Postal address: DIAM, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. Email address: a.c.fey-denboer@tudelft.nl
∗∗ Postal address: Department of Mathematics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: rhofstad@win.tue.nl
∗∗∗ Current address: ORTEC BV, Orlyplein 145c, 1043 DV Amsterdam, The Netherlands. Email address: mklok@ortec.nl
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Abstract

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We study sample covariance matrices of the form W = (1 / n)C C T, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or k n → ∞ with k n = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C . This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

Keywords

Sample covariance matrixlarge deviationeigenvalueCDMA with soft-decision parallel interference cancellation

MSC classification

Primary: 60F10: Large deviations 62H20: Measures of association (correlation, canonical correlation, etc.) 94A99: None of the above, but in this section

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1048 - 1071
Copyright
Copyright © Applied Probability Trust 2008 

References

Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Prob. 26, 316–345.Google Scholar
Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Prob. 32, 553–605.Google Scholar
Bai, Z. D. and Yin, Y. Q. (1993). Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix. Ann. Prob. 21, 1275–1294.CrossRefGoogle Scholar
Bai, Z. D., Silverstein, J. W. and Yin, Y. Q. (1988). A note on the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivariate Anal. 26, 166–168.Google Scholar
Buehrer, R. M., Nicoloso, S. P. and Gollamudi, S. (1999). Linear versus non-linear interference cancellation. J. Commun. Networks 1, 118–133.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
Den Hollander, F. (2000). Large Deviations (Fields Inst. Monogr. 14). American Mathematical Society, Providence, RI.Google Scholar
Fey-den Boer, A. C., van der Hofstad, R. and Klok, M. J. (2003). Linear interference cancellation in CDMA systems and large deviations of the correlation matrix eigenvalues. In Proc. Symp. IEEE Benelux Chapter Commum. Veh. Tech. (Eindhoven, 2003).Google Scholar
Grant, A. and Schlegel, C. (2001). Convergence of linear interference cancellation multiuser receivers. IEEE Trans. Commun. 49, 1824–1834.Google Scholar
Guionnet, A. (2002). Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188, 461–515.Google Scholar
Guo, D., Rasmussen, L. K., Tun, S. and Lim, T. J. (2000). A matrix-algebraic approach to linear parallel interference cancellation in CDMA. IEEE Trans. Commun. 48, 152–161.Google Scholar
Hiai, F. and Petz, D. (1998). Eigenvalue density of the Wishart matrix and large deviations. Infin. Dimens. Anal. Quantum Prob. 1, 633–646.Google Scholar
Klok, M. J. (2001). Performance analysis of advanced third generation receivers. Doctoral Thesis, Delft University of Technology.Google Scholar
Marchenko, V. A. and Pastur, L. A. (1967). The distribution of eigenvalues in certain sets of random matrices. Mat. Sb. 72, 507–536.Google Scholar
Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability (Encyclopedia Math. Appl. 100). Cambridge University Press.Google Scholar
Rasmussen, L. K. and Oppermann, I. J. (2003). Ping-pong effects in linear parallel interference cancellation for CDMA. IEEE Trans. Wireless Commun. 2, 357–363.Google Scholar
Rogers, C. A. (1963). Covering a sphere with spheres. Mathematika 10, 157–164.Google Scholar
Van der Hofstad, R. and Klok, M. J. (2003). Performance of DS-CDMA systems with optimal hard-decision parallel interference cancellation. IEEE Trans. Inf. Theory 25, 2918–2940.CrossRefGoogle Scholar
Van der Hofstad, R. and Klok, M. J. (2004). Improving the performance of third-generation wireless communication systems. Adv. Appl. Prob. 36, 1046–1084.Google Scholar
Van der Hofstad, R., Hooghiemstra, G. and Klok, M. J. (2002). Analytical methods for CDMA systems with parallel interference cancellation: the large deviation approach. Wireless Personal Commun. 21, 289–307.Google Scholar
Van der Hofstad, R., Hooghiemstra, G. and Klok, M. J. (2002). Large deviations for code division multiple access systems. SIAM J. Appl. Math. 62, 1044–1065.Google Scholar
Van der Hofstad, R., Löwe, M. and Vermet, F. (2006). The effect of system load on the existence of bit errors in CDMA with and without parallel interference cancelation. IEEE Trans. Inf. Theory 52, 4733–4741.Google Scholar
Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Prob. Theory Relat. Fields 78, 509–521.CrossRefGoogle Scholar