Let $N,X_1 ,X_2 , \cdots ,$ be independent random variables with $X_1 ,X_2 , \cdots ,$ being nonnegative and identically distributed. Let N have a power series distribution. Considering the random sum $S = \sum _{i = 1}^N X_i$, the present paper gives a characterization of the distributions of N and $X_i $ by means of the property that, up to a scale parameter, S has the same distribution as $X_i $ . If the expectation of $X_i $ is finite, one obtains a characterization of the gamma distribution.