Let $ {\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ be a sequence random variables
For the summation of those random variable:
$ {\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}}$
We know if each $X_i$ is a random variable for a independent simple binominal random walk, then ${\bar {X}}_{n}$ follow a normal distribution.
Question: What kind of simple sequences of random variables can naturally lead to a Gamma distribution (like binormal random walks do for normal distribution) ?
Best Answer
Let $(X_k)_{k\leq n}$ be IID s.t. $X_1 \sim \textrm{Exp}(\lambda)$. Then $\overline{X}_{n}\sim \textrm{Gamma}(n,\lambda n)$; with shape-rate parametrization. To see this: $$\begin{aligned}E[e^{i\xi(X_1+....+X_n)/n}]&=(E[e^{i\xi X_1/n}])^n=\\ &=\bigg(\frac{\lambda}{\lambda-i\xi/n}\bigg)^n=\\ &=(1-i\xi/(\lambda n))^{-n}\end{aligned}$$ The last line is the characteristic function of the $\textrm{Gamma}(n,\lambda n)$ law. The assertion follows from $\{x\mapsto e^{i\xi x}:\xi \in \mathbb{R}\}$ being a determining class on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$.