Central limit theorem of random quadratics forms involving random matrices☆
Introduction
Let random variables be independent and identically distributed (i.i.d.) with and . Define , , , . Let be the eigenvalues of and be the empirical spectral distribution of , i.e., It is well known that if and then where denotes the limiting spectral distribution. There are many literatures about the convergence of (see Bai, 1999 and the references given there).
Lately, the theory of large dimensional random matrices has been a popular tool in the research of the asymptotic problem of code division multiple-access (CDMA) communication systems. For example, one may refer to the work of Tse and Hanly (1999) and Tse and Zeitouni (2000). The application in the communication also puts forward challenges to the large dimensional matrix theory. For instance, to analyze the large system performance of the signal-to-inference (SIR) of some kind of receiver, the random quadratic forms involving random matrix, and , were introduced by Trichard et al. (2002). However, only the convergence in probability for a special random sequence was given in their work. As we know, the central limit theorem is more powerful than the convergence in probability. Even in wireless communications, central limit theorem can also be used to characterize the fluctuation of the performance of SIR around the asymptotic value. The main object of this paper is to prove that is asymptotic normal under the assumption . Here we would like to emphasize that the random vector is not independent of the random matrix .
Section snippets
Main results
For convenience, some notations are firstly introduced. Let where is the limiting spectral distribution, is a Brownian bridge independent of , which is . Here is obtained from by replacing y with and other analogues below such as have similar meanings. Moreover, is required in the definition of . Our main results are as
The proof of theorem
To simplify the notations, the constant c may denote different values at different occasions in what follows. Before proceeding, we first need some lemmas. Lemma 3.1 Let be a function of the random vector and be a random variable. Suppose that for , where and both are the non-random distribution functions, and denote the sets of the continuity points of and , respectively. Then
Acknowledgment
The authors would like to thank an anonymous referee for his constructive comments.
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Research supported by Grant 10471135 and 10571001 from the National Natural Science Foundation of China.