The KLR-Theorem Revisited
- 4 Downloads
For independent random variables X1,…,Xn;Y1,…,Yn with all Xi identically distributed and same for Yj, we study the relation
with a,b some constants. It is proved that for n ≥ 3 and ab > 0 the relation holds iff Xi and Yj are Gaussian. A new characterization arises in case of a = 1,b = − 1. In this case either Xi or Yj or both have a Gaussian component. It is the first (at least known to the author) case when presence of a Gaussian component is a characteristic property.
Keywords and phrases.Constancy of regression-Characterization Gaussian component
AMS (2000) subject classification.Primary 62H05 Secondary 62J02
My collaboration with C. R. Rao goes back to 1965 when I was at the Steklov Mathematical Institute in Leningrad and Rao was at the Indian Statistical Institute in Calcutta, and lasted for some 40 years. It gives me a special pleasure to dedicate this paper to C. R. Rao on the occasion of his centenary.
- Zinger, A.A. and Linnik, Y.V. (1955). On an analytic generalization of the Cramer theorem in Russian). Vestnik of Leningrad Univ.11, 51–56.Google Scholar
- Kagan, A.M. and Klebanov, L.B. (2010). A class of multivariate distributions related to distributions with a Gaussian component. In: IMS collections, 7, 105–112.Google Scholar