论文标题
关于Marcenko-Pastur问题的概括
On a Generalisation of the Marcenko-Pastur Problem
论文作者
论文摘要
我们研究了广义的WishArt矩阵的频谱，定义为$ \ MathBf {f} =（x y^\ top + y x^\ top）/2t $，其中$ x $和$ y $是$ n \ times t $ t $ hatrices，其零是零是零的，单位方差iid条目以及这样的$ \ mathbbbb and ymathbbbb {e} δ_{i，j} $。限制$ C = 1 $对应于Marcenko-Pastur问题。对于一般的$ c $，我们表明$ \ mathbf {f} $的stietjes变换是立方方程的解决方案。在限制中，$ c = 0 $，$ t \ gg n $特征值的密度收敛到Wigner半圆。
We study the spectrum of generalized Wishart matrices, defined as $\mathbf{F}=( X Y^\top + Y X^\top)/2T$, where $X$ and $Y$ are $N \times T$ matrices with zero mean, unit variance IID entries and such that $\mathbb{E}[X_{it} Y_{jt}]=c δ_{i,j}$. The limit $c=1$ corresponds to the Marcenko-Pastur problem. For a general $c$, we show that the Stietjes transform of $\mathbf{F}$ is the solution of a cubic equation. In the limit $c=0$, $T \gg N$ the density of eigenvalues converges to the Wigner semi-circle.
