The KLR-theorem revisited

2019/02/18 by Kagan, Abram M.
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1902.06800

Abstract

For independent random variables X₁,…, X_n;Y₁,…, Y_n with all X_i identically distributed and same for Y_j, we study the relation E\a X + b Y|X₁ - X +Y₁ - Y,…,X_n - X +Y_n - Y\=\rm const with a, b some constants. It is proved that for n≥ 3 and ab>0 the relation holds iff X_i and Y_j are Gaussian. A new characterization arises in case of a=1, b= -1. In this case either X_i or Y_j 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.

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