On the stability of one characterization of the stable distributions

Abstract

As early as 1923, Georg P6lya wrote: ``The Gaussian error law possesses the property that it remains valid under a linear combination of errors. The Gaussian error law can be characterized by this property to some extent – it is the only law that admits steadiness with respect to linear combinations of errors''. The idea of using linear combinations of random variables to cha­racterize the stable distributions has been extended by P. Levy. We investigate the stability of this characterization.