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The KLR-Theorem Revisited

Author

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  • Abram Kagan

    (University of Maryland)

Abstract

For independent random variables X1,…,Xn;Y1,…,Yn with all Xi identically distributed and same for Yj, we study the relation E { a X ̄ + b Y ̄ | X 1 − X ̄ + Y 1 − Y ̄ , … , X n − X ̄ + Y n − Y ̄ } = const $$ E\{a\bar X + b\bar Y|X_{1} -\bar X +Y_{1} -\bar Y,\ldots,X_{n} -\bar X +Y_{n} -\bar Y\}=\text{const} $$ 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.

Suggested Citation

  • Abram Kagan, 2021. "The KLR-Theorem Revisited," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 549-553, August.
    • Handle: RePEc:spr:sankha:v:83:y:2021:i:2:d:10.1007_s13171-019-00183-2
    DOI: 10.1007/s13171-019-00183-2
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