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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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