Asymptotically chi-squared?

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1 total messages Started by ALEWAND Tue, 21 Jul 1992 15:38

#58

Author: ALEWAND
Date: Tue, 21 Jul 1992 15:38

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Let X1,X2,.. be a sequence of k-dimensional random vectors (k>1),
t1,t2,... a sequence of real vectors with    lim tn=0,  and
        sqrt(n) (Xn-tn)
converges in distribution to the multinormal distribution  N(0,SIGMA).
 
SIGMA is idempotent with rank(SIGMA)=s and 1<=s<=k-1.
 
My conjecture is:
 
n (Transpose(Xn) Xn -Transpose(tn) tn)  is asymptotically chi-squared
with s degrees of freedom if and only if
 
Lim n Transpose(tn) tn =0  (n to infinity).
 
Is anybody able to prove (or disprove) this conjecture?
 
Thanks
 
Achim Lewandowski
Institut fuer Statistik und Oekonometrie
3400 Goettingen
Germany
email: alewand@ibm.gwdg.de

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