Multivariate Versions of Cochran′s Theorems II

✍ Scribed by C.S. Wong; T.H. Wang

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 336 KB
• Volume: 44
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1993.1008

✦ Synopsis

A general easily checkable Cochran theorem is obtained for a normal random operator (Y). This result does not require that the covariance, (\Sigma_{\mathbf{r}}), of (Y) is nonsingular or is of the usual form (A \otimes 2); nor does it assume that the mean. (\mu). of (Y) is equal to zero. Indeed, (\left{Y^{\prime} W_{i} Y\right}) (with nonnegative definite (W_{i}) s) is a family of independent Wishart random operators (Y^{\prime} W_{i} Y^{\prime}) of parameter (\left(m_{i}, \Sigma_{1}, i_{1}\right)) if and only if for some nonnegative definite (A) and for all (i \neq j) : (a) (\left(W_{i} \otimes /\right)\left(2_{1}-A \otimes 2\right)\left(W_{1} \otimes I\right)=0 ;) (b) (A W_{1} A W_{i}=A W_{i}, r\left(A W_{i}\right)=m_{1}, \quad) (c) (i_{1}=) (\mu^{\prime} W_{i} \mu=\mu^{\prime} W_{i} A W_{i} \mu); and (d) (\left(W_{i} \otimes l\right) \Sigma_{\gamma}\left(W_{i} \otimes l\right)=0). The usual multivariate versions of Cochran's theorem are contained in a special case of our result where (\Sigma_{1}=A \otimes 2). The (A) in our version of Cochran's theorem can actually be constructed from (\Sigma, \Sigma_{r}), and the sum of the (W_{i}^{\prime}) 's. ' 1993 Academic Press, tne.