Sankhya A

pp 1–5 | Cite as

The KLR-Theorem Revisited

  • Abram KaganEmail author


For independent random variables X1,…,Xn;Y1,…,Yn with all Xi identically distributed and same for Yj, we study the relation

$$ E\{a\bar X + b\bar Y|X_{1} -\bar X +Y_{1} -\bar Y,\ldots,X_{n} -\bar X +Y_{n} -\bar Y\}=\text{const} $$

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.


  1. Lukacs, E. (1942). A characterization of he normal distribution. Ann. Math. Stat.13, 91–93.CrossRefGoogle Scholar
  2. 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
  3. Kagan, A.M., Linnik, Y.V. and Rao, C.R. (1965). On a characterization of the normal law based on a property of the sample average. SankhyaA 27, 405–406.MathSciNetzbMATHGoogle Scholar
  4. Kagan, A.M., Linnik, Y.V. and Rao, C.R. (1973). Characterization problems in mathematical statistics. Wiley, New York.zbMATHGoogle Scholar
  5. 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

Copyright information

© Indian Statistical Institute 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations