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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: [email protected]
∗∗ Postal address: Department of Mathematics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗∗ Current address: ORTEC BV, Orlyplein 145c, 1043 DV Amsterdam, The Netherlands. Email address: [email protected]
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Abstract

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We study sample covariance matrices of the form W = (1 / n)CCT, 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 kn → ∞ with kn = 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
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1048 - 1071
Copyright
Copyright © Applied Probability Trust 2008 

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