We consider a real random vector X with cumulant generating function Psi, non-singular covariance matrix and zero expectation and prove that a principal component transformation of X is the Jacobian at the origin of a Morse-transformation for Psi. Subsequently, a similar result is stated removing the nondegeneracy assumption. Finally we discuss the possibility of applying our results in a statistical framework.
A note on principal components and Morse Theorem
CAROTA, Cinzia;
1996-01-01
Abstract
We consider a real random vector X with cumulant generating function Psi, non-singular covariance matrix and zero expectation and prove that a principal component transformation of X is the Jacobian at the origin of a Morse-transformation for Psi. Subsequently, a similar result is stated removing the nondegeneracy assumption. Finally we discuss the possibility of applying our results in a statistical framework.
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