Which random variable sequences can naturally generate gamma distributions? 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) ?
 A: 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 $(\mathbb{R},\mathscr{B}(\mathbb{R}))$.
A: We can generalize the previous answer.
Let $X_i \sim \Gamma(k_i, \theta)$ be independent (notice that the shape parameter can be different while the scale one is assumed to be the same). Then
$$
X := \sum_{i=1}^{n}X_i \sim \Gamma\bigg(\sum_{i=1}^{n}k_i, \theta \bigg)
$$
Hence, multiplying by $1/n$ and remembering the scaling property of a gamma distribution, we have that $X \sim \Gamma\bigg(\sum_{i=1}^{n}k_i, \frac{\theta}{n} \bigg)$.
This means that the sum of independent gamma distribution generates a gamma distribution. So, as special cases, we have that the sum of exponential distribution and the sum of independent chi-square distribution are gamma distributed.
