A Simple Proof of a Classical Theorem which Characterizes the Gamma Distribution
The Annals of Statistics · 1978 · Volume 6 · Issue 5
Abstract
The following result of Lukacs is known: let $X_1, X_2$ be independent, positive random variables, having the nondegenerate distributions $P_1$ and $P_2$. Suppose that $X_1/X_2$ and $X_1 + X_2$ are independent. Then $P_1$ and $P_2$ are gamma distributions with the same scale parameter. Lukacs' original deduction requires details from complex analysis. Here a simpler proof is given. Instead of $P_1$ and $P_2$ two other probability measures $\mu_1$ and $\mu_2$ are shown to be determined by the independence properties of $X_1$ and $X_2$. It is possible to express $P_i$ and $\mu_i$ by each other, and $\mu_i$ is chosen such that all moments of $\mu_i$ are finite $(i = 1,2)$. Thus the proof reduces to a straight-forward calculation of moments.
Keywords
62E10, 62H05, Characteristic properties of distributions, didactical revisions of known deductions, gamma distribution
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