Weak convergence of random functions defined by the eigenvectors of sample covariance matrices
Let \(V_ n=(v_{ij})\) be an \(n\times s(n)\) matrix, where \(\lim_{n\to \infty}n/s(n)>0\) and \(\{v_{ij}\); i,j\(\geq 1\}\) is an array of i.i.d. random variables. Assume that \(O_ n\Lambda_ nO^ T_ n\) is the spectral decomposition of the matrix \(M_ n=(1/s(n))V_ nV^ T_ n.\) For \(x_ n\in R^ n\) with \(\| x_ n\| =1\), define \((y_ 1,...,y_ n)=O^ T_ nx_ n\) and then \[ X_ n(t)=(n/2)^{1/2}\sum (y^ 2_ i-n^{-1}) \] for each \(t\in (0,1)\), where the sum is taken over all \(i\leq nt\). Conditions are given under which \(X_ n\) converges weakly in D[0,1] to a Brownian bridge. The matrix \(M_ n\) can be considered to be a sample covariance matrix, and the result explores the limiting behaviour of the eigenvectors of \(M_ n\).
eigenvectors of sample covariance matrix
scientific article; zbMATH DE number 4163942