论文标题
由大维随机矩阵的特征向量定义的随机函数集合的弱收敛性
Weak Convergence of a Collection of Random Functions Defined by the Eigenvectors of Large Dimensional Random Matrices
论文作者
论文摘要
对于每个$ n $，让$ u_n $是haar分发在$ n \ times n $统一矩阵上。令$ \ bfx_ {n，1}，\ ldots，\ bfx_ {n，m} $表示正交nonrandom单位向量$ {\ bbb c}^n $，然后$ \ text {\ bf {\ bf {\ bf u} _ {n，k} =（u_k^1，\ ldots，u_k^n）^*= u^*\ text {\ bf x} _ {n，k} $，$ k = 1，\ ldots，m $。在[0,1]：$ x^{k，k} _n（t）= \ sqrt n \ sum_ {i = 1}^{[nt]}（| u_k^i |^i |^2- \ tfrac1n）$，美元％（“ $ \ bar {\，\，\，\，\，} $”表示共轭）。然后证明，$ x_n^{k，k}，\ re x_n^{k，k'} $，$ \ im x_n^{k，k'} $，在$ d [0,1] $中被视为随机过程，薄弱地收敛，作为$ n \ to $ n \ to \ n \ to \ m^$ m^$ m^2 $独立oprynian brown of brown of brown of brown of布朗桥。 在实际情况下，对于$ m（m+1）/2 $过程的相同结果，其中$ o_n $是真实的正交haar分发的，$ \ bfx_ {n，i} \ in {\ bbb r}^n $，带有$ \ sqrt n $ in $ \ sqrt n $ in $ x^n $ in $ x^n $ $ x_n^{k，k'} $分别替换为$ \ sqrt {\ frac n2} $和$ \ sqrt {n} $。后一个结果将显示为$ m_n =（1/s）v_nv_n^t $的eigenVectors矩阵中，其中$ v_n $是$ n \ times s $由$ \ {v_ {ij} \}的条目组成的$ n \ times s $，标准化和对称分布，每个$ \ bfx_ {n，i} = \ {\ pm1/\ sqrt n，\ ldots，\ pm1/\ sqrt n \} $，以及$ n/s \ to y> 0 $ n/s \ to y> 0 $ n \ as $ n \ to \ n \ to \ infty $。该结果扩展了J.W. Silverstein {\ Sl Ann。概率。 \ bf18} 1174-1194。 这些结果应用于检测问题，在采样随机矢量中主要由噪声制成，并检测样品是否包括非随机矢量。
For each $n$, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $\bfx_{n,1},\ldots,\bfx_{n,m} $ denote orthogonal nonrandom unit vectors in ${\Bbb C}^n$ and let $\text{\bf u}_{n,k}=(u_k^1,\ldots,u_k^n)^*=U^*\text{\bf x}_{n,k}$, $k=1,\ldots,m$. Define the following functions on [0,1]: $X^{k,k}_n(t)=\sqrt n\sum_{i=1}^{[nt]}(|u_k^i|^2-\tfrac1n)$, $X_n^{k,k'}(t)=\sqrt{2n}\sum_{i=1}^{[nt]}\bar u_k^iu_{k'}^i$, $k<k'$. %("$\bar{\,\,\,\,\,}$" denoting conjugate). Then it is proven that $X_n^{k,k},\Re X_n^{k,k'}$, $\Im X_n^{k,k'}$, considered as random processes in $D[0,1]$, converge weakly, as $n\to\infty$, to $m^2$ independent copies of Brownian bridge. The same result holds for the $m(m+1)/2$ processes in the real case, where $O_n$ is real orthogonal Haar distributed and $\bfx_{n,i}\in{\Bbb R}^n$, with $\sqrt n$ in $X^{k,k}_n$ and $\sqrt{2n}$ in $X_n^{k,k'}$ replaced with $\sqrt{\frac n2}$ and $\sqrt{n}$, respectively. This latter result will be shown to hold for the matrix of eigenvectors of $M_n=(1/s)V_nV_n^T$ where $V_n$ is $n\times s$ consisting of the entries of $\{v_{ij}\},\ i,j=1,2,\ldots$, i.i.d. standardized and symmetrically distributed, with each $\bfx_{n,i}=\{\pm1/\sqrt n,\ldots,\pm1/\sqrt n\}$, and $n/s\to y>0$ as $n\to\infty$. This result extends the result in J.W. Silverstein {\sl Ann. Probab. \bf18} 1174-1194. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector.
