A Characterization of Uniform Distribution

Joanna Chachulska¹

¹ Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 2, pp. 207-220.

Résumé

Is the Lebesgue measure on $[0,1]^2$ a unique product measure on $[0,1]^2$ which is transformed again into a product measure on $[0,1]^2$ by the mapping $\psi(x,y)=(x,(x+y)\bmod 1))$? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables $X$ and $Y$ constancy of the conditional expectations of $X+Y-I(X+Y>1)$ and its square given $X$ identifies uniform distribution either absolutely continuous or discrete. No assumptions are imposed on the supports of the distributions of $X$ and $Y$.

Détail

DOI : 10.4064/ba53-2-9

Keywords: lebesgue measure unique product measure which transformed again product measure mapping psi bmod here somewhat stronger version problem probabilistic framework answered shown independent identically distributed random variables constancy conditional expectations y i its square given identifies uniform distribution either absolutely continuous discrete assumptions imposed supports distributions