论文标题
大尺寸一般信息加上噪声型矩阵的限制光谱分布
The limiting spectral distribution of large dimensional general information-plus-noise type matrices
论文作者
论文摘要
令$ x_ {n} $ be $ n \ times n $随机复杂矩阵，$ r_ {n} $和$ t_ {n} $分别为dimensions $ n \ times n $和$ n \ times n $的非随机复杂矩阵。我们假设$ x_ {n} $的条目是独立且分布相同的，$ t_ {n} $是无负的确定性遗传学矩阵和$ t_ {n} r_ {n} r_ {n} r_ {n}^{n}^{*} = r_ {n} n} r_ {n} r_ {n} r_ {n} r_ {n} { 一般信息及 - noise类型矩阵由$ c_ {n} = \ frac {1} {n} {n} t_ {n}^{\ frac {\ frac {1} {2}}}} \ left（r_ {n} +x_ {n} +x_ {n} \ right） \ left（r_ {n}+x_ {n} \ right）^{*} t_ {n}^{\ frac {1} {2}}} $。 在本文中，我们建立了大尺寸一般信息加上噪声类型矩阵$ c_ {n} $的极限光谱分布。具体来说，我们表明，作为$ n $和$ n $倾向于成比例地，$ c_ {n} $的特征值的经验分布薄弱地收敛到非随机概率分布，这是根据其stieltjes变换方程式的特征。
Let $ X_{n} $ be $ n\times N $ random complex matrices, $R_{n}$ and $T_{n}$ be non-random complex matrices with dimensions $n\times N$ and $n\times n$, respectively. We assume that the entries of $ X_{n} $ are independent and identically distributed, $ T_{n} $ are nonnegative definite Hermitian matrices and $T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} $. The general information-plus-noise type matrices are defined by $C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\right) \left(R_{n}+X_{n}\right)^{*}T_{n}^{\frac{1}{2}} $. In this paper, we establish the limiting spectral distribution of the large dimensional general information-plus-noise type matrices $C_{n}$. Specifically, we show that as $n$ and $N$ tend to infinity proportionally, the empirical distribution of the eigenvalues of $C_{n}$ converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform.
