Some Intrinsic Properties of the Gamma Distribution

Let {Yn} be a sequence of nonnegative random variables (rvs), and Sn = ∑n j=1 Yj , n ≥ 1. It is first shown that independence of Sk−1 and Yk, for all 2 ≤ k ≤ n, does not imply the independence of Y1, Y2, . . . , Yn. When Yj ’s are identically distributed exponential Exp(α) variables, we show that the independence of Sk−1 and Yk, 2 ≤ k ≤ n, implies that the Sk follows a gamma G(α, k) distribution for every 1 ≤ k ≤ n. It is shown by a counterexample that the converse is not true. We show that if X is a non-negative integer valued rv, then there exists, under certain conditions, a rv Y ≥ 0 such that N(Y ) L = X, where {N(t)} is a standard (homogeneous) Poisson process, and obtain the Laplace-Stieltjes transform of Y . This leads to a new characterization for the gamma distribution. It is also shown that a G(α, k) distribution may arise as the distribution of Sk, where the components are not necessarily exponential. Several typical examples are discussed.