Some Characterizations of the Exponential Distribution by Geometric Compounding

Abstract

If, for every $p \in ( {0,1} ),p$ times a geometric $( p )$ sum of independent identically distributed nonnegative random variables has the same distribution as the individual random variables, then the random variables are exponentially distributed. The phase “for every $p \in ( {0,1} )$ ” can be weakenedsomewhat, and if first moments are assumed to exist, it can be replaced by “for some $p \in ( {0,1} )$.” Related characterizations of the power function distribution and of the Poisson process are discussed, as are the effects of dropping the assumption of nonnegativity of the random variables.