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The asymptotic distribution of singular values with applications to canonical correlations and correspondence analysis. (English) Zbl 0805.62020
Let \(X_ n\), \(n=1,2,\dots\) be a sequence of \(p\times q\) random matrices, \(p\geq q\). Assume that there exists a fixed \(p\times q\) matrix \(B\) and a sequence of real constants \(b_ n\to\infty\) such that the random matrix \(Z_ n= b_ n (X_ n -B)\) converges in distribution to a limit matrix \(Z\). Let \(\psi(X_ n)\) (resp. \(\psi(B)\)) denote the \(q\)-vector of singular values of \(X\) (resp. \(B\)).
The main result of the author describes the limiting distribution of \(E_ n= b_ n (\psi(X_ n)- \psi(B))\) as a function of \(B\) and of the limit matrix \(Z\). Typically, the limiting distribution can be normal when \(Z\) is multivariate normal and the singular values of \(b\) are all strictly positive and of multiplicity one. The primary tool for proving the main result is a perturbation inequality for singular values. Applications to canonical correlations and correspondence analysis are also given.

MSC:
62E20 Asymptotic distribution theory in statistics
62H25 Factor analysis and principal components; correspondence analysis
62H20 Measures of association (correlation, canonical correlation, etc.)
15A42 Inequalities involving eigenvalues and eigenvectors
15A18 Eigenvalues, singular values, and eigenvectors
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