arXiv math/04113332004-11math.PRmath.STstat.TH

给定方差分布的Gram矩阵特征值的经验分布

The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile

W. Hachem, P. Loubaton, J. Najim

备注：: 25 pages, revised version. Assumption (A2) has been relaxed

摘要：

考虑$N\times n$随机矩阵$Y_n=(Y_{ij}^{n})$，其中条目由$Y_{ij}^{n}=\frac{σ(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$给出，$X_{ij}^{n}$在i.i.d.并且$σ:[0,1]^2 \to (0,\infty)$是称为方差分布的连续函数。现在考虑非对角元素为零的确定性$N\times n$矩阵$Λ_n=(Λ_{ij}^{n})$。用$Σ_n$表示非中心矩阵$Y_n + Λ_n$。然后在$\lim_{n\to \infty} \frac Nn =c>0$和$$ \frac{1}{N} \sum_{i=1}^{N} δ_{(\frac{i}{N}, (Λ_{ii}^n)^2)} \xrightarrow[n\to \infty]{} H(dx,dλ), $$的假设下，证明了$ Σ_n Σ_n^T$的特征值的经验分布在分布上几乎必然收敛于一个非随机的概率测度.这种措施的特点是在其Stieltjes变换，这是得到一个辅助系统的方程的帮助。这种结果在无线通信领域引起了人们的兴趣。

英文摘要：

Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=\frac{σ(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the $X_{ij}^{n}$ being centered i.i.d. and $σ:[0,1]^2 \to (0,\infty)$ being a continuous function called a variance profile. Consider now a deterministic $N\times n$ matrix $Λ_n=(Λ_{ij}^{n})$ whose non diagonal elements are zero. Denote by $Σ_n$ the non-centered matrix $Y_n + Λ_n$. Then under the assumption that $\lim_{n\to \infty} \frac Nn =c>0$ and $$ \frac{1}{N} \sum_{i=1}^{N} δ_{(\frac{i}{N}, (Λ_{ii}^n)^2)} \xrightarrow[n\to \infty]{} H(dx,dλ), $$ where $H$ is a probability measure, it is proven that the empirical distribution of the eigenvalues of $ Σ_n Σ_n^T$ converges almost surely in distribution to a non random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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