Central limit theorem of random quadratics forms involving random matrices

Abstract

Let $s_i=1/\sqrt{N}(v_{i,1},\dots ,v_{i,N}{)}^{\mathrm{T}}$ and $S=(s_1,s_2,\dots ,s_K)$ where random variables $\{v_{\mathit{ij}},i,j=1,2,\dots ,\}$ are i.i.d. with ${\mathit{Ev}}_{11}^4<\infty$. The central limit theorem of the random quadratic forms $s_1^{\mathrm{T}}({\mathit{SS}}^{\mathrm{T}}{)}^i{s}_1$ is established, which arises from the application in wireless communications.

Introduction

Let random variables $\{v_{\mathit{ij}},i,j=1,\dots \}$ be independent and identically distributed (i.i.d.) with ${\mathit{Ev}}_{11}=0$ and ${\mathit{Ev}}_{11}^2=1$. Define $s_i=1/\sqrt{N}(v_{i,1},\dots ,v_{i,N}{)}^{\mathrm{T}}$, $i=1,\dots ,K$, $S_1=(s_2,s_3,\dots ,s_K)$, $S=(s_1,s_2,\dots ,s_K)$. Let ${\lambda }_1⩽{\lambda }_2⩽\cdots ⩽{\lambda }_N$ be the eigenvalues of $(N/K)S_1{S}_1^{\mathrm{T}}$ and $F_N$ be the empirical spectral distribution of $(N/K)S_1{S}_1^{\mathrm{T}}$, i.e.,

$$F_N(x)=\frac{{\sum }_{i=1}^{N}I({\lambda }_i⩽x)}{N}\text{.}$$

It is well known that if ${\mathit{Ev}}_{11}^2<\infty$ and ${\lim }_{N\to \infty }K/N=y>0,$ then

$$F_N(x)\stackrel{\mathrm{a}.\mathrm{s}.}{⟶}{F}_y(x)\text{,}$$

where $F_y(x)$ denotes the limiting spectral distribution. There are many literatures about the convergence of $F_N(x)$ (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, $s_1^{\mathrm{t}}(S_1{S}_1^{\mathrm{T}}{)}^i{s}_1$ and $s_1^{\mathrm{t}}({\mathit{SS}}^{\mathrm{T}}{)}^i{s}_1$, 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 $s_1^{\mathrm{t}}({\mathit{SS}}^{\mathrm{T}}{)}^i{s}_1$ is asymptotic normal under the assumption ${\mathit{Ev}}_{11}^4<\infty$. Here we would like to emphasize that the random vector $s_1$ is not independent of the random matrix ${\text{<mi mathvariant="italic" is="true">SS</mi>}}^{\mathrm{T}}$.