A probabilistic note on the Cauchy functional equation

  • Sergey N. SmirnovEmail author


In this short note we show that a weak version of Bernstein’s characterization of the normal distribution implies the local integrability of a measurable solution of the Cauchy functional equation; the linearity of a solution of the Cauchy functional equation is an easy consequence of its local integrability. In its turn, this weak version of Bernstein’s theorem can be derived from Cauchy’s theorem.


Cauchy functional equation Measurability Local integrability Bernstein’s characterization theorem Normal distribution Characteristic function 

Mathematics Subject Classification

39B22 60E10 62E10 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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