一种新的行列相关随机矩阵模型的极限谱分布
Limiting spectral distribution of a new random matrix model with dependence across rows and columns
摘要：
我们引入了一个随机矩阵模型，其中的条目在行和列上都是相关的。更确切地说，我们调查矩阵的形式$\X=(X_{(i-1)n+t})_{it}\in\R^{p\times n}$来自一个线性过程$X_t=\sum_j c_j Z_{t-j}$，其中的$\{Z_t\}$是独立的随机变量有界的四阶矩。我们表明，当$p$和$n$趋于无穷大，使比率$p/n$收敛到一个有限的正极限$y$，经验谱分布的$p^{-1}\X\X^{\T}$几乎肯定收敛到一个确定性的措施。这种限制措施，这取决于$y$和线性过程$X_t$的谱密度，其特征在于由一个积分方程的Stieltjes变换。矩阵$p^{-1}\X\X^{\T}$可以解释为一个高维过程的样本协方差矩阵的近似值，该过程的分量是$X_t$的独立副本。
英文摘要：
We introduce a random matrix model where the entries are dependent across both rows and columns. More precisely, we investigate matrices of the form $\X=(X_{(i-1)n+t})_{it}\in\R^{p\times n}$ derived from a linear process $X_t=\sum_j c_j Z_{t-j}$, where the $\{Z_t\}$ are independent random variables with bounded fourth moments. We show that, when both $p$ and $n$ tend to infinity such that the ratio $p/n$ converges to a finite positive limit $y$, the empirical spectral distribution of $p^{-1}\X\X^{\T}$ converges almost surely to a deterministic measure. This limiting measure, which depends on $y$ and the spectral density of the linear process $X_t$, is characterized by an integral equation for its Stieltjes transform. The matrix $p^{-1}\X\X^{\T}$ can be interpreted as an approximation to the sample covariance matrix of a high-dimensional process whose components are independent copies of $X_t$.
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