Cauchy Equation on Discrete Domain and Some Characterization Theorems

Abstract

Discrete version of normal distribution, i.e., $P(x) = c\exp \{ - \beta x^2 \} , \beta > 0, x \in {\bf Z}$, is characterized via the solution of Cauchy type equation on discrete domain in dimension 4 or higher. It is also shown that this characterization does not necessarily hold for second and third dimensions. Some statistical aspects of radial symmetry and eccentricity along with the properties of this distribution are also discussed.