Cauchy equation on discrete domain and some characterisation theorems

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