Some Intrinsic Properties of the Gamma Distribution

抄録

Let \{Y_n\} be a sequence of nonnegative random variables (rvs), and S_n=∑_j=1ⁿY_j, n≥1. It is first shown that independence of S_k-1 and Y_k, for all 2≤ k≤n, does not imply the independence of Y₁,Y₂,...,Y_n. When Y_j's are identically distributed exponential \Exp(α) variables, we show that the independence of S_k-1 and Y_k, 2W≤k≤n, implies that the S_k 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)\stackrel{\cal{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 S_k, where the components are not necessarily exponential. Several typical examples are discussed.