Sum of independent Gamma distributions is a Gamma or a normal distribution? Let $ {\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}$ be a sequence of independent random variables, each of those random variable follow a Gamma distribution.
For the summation of those random variable:
$ {\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}}$
Question: Is the summation of independent Gamma distributions a Gamma distribution or a normal distribution ?
Here is the confusion: the summation of Gamma distribution should still be a Gamma distribution. On the other hand, central limit theorem says that such summation should approach a normal distribution. which of those viewpoint is correct ?
 A: The answer depends on whether the gamma distributions have the same rate parameter.
if $X_i \sim \operatorname{Gamma}(a_i, b)$ then their sample total will also be gamma distributed:  $$\sum X_i \sim \operatorname{Gamma}\left(\sum a_i, b \right),$$ hence the sample mean will also be gamma:  $$\bar X \sim \operatorname{Gamma}\left(\sum a_i, nb\right);$$ that is to say, $\bar X$ is gamma with shape equal to the sum of shapes, and rate equal to $n$ times the original rate.
However, if the rate parameter is not the same for each $X_i$, then in general, the sample mean will not be gamma distributed.  There is no general closed form.
That said, the asymptotic behavior of the sample mean does obey the central limit theorem.  The exact distribution in the common rate case above is indeed gamma; as $n \to \infty$, such a gamma distribution tends to a normal distribution with mean $\mu = \bar a/b$ where $\bar a = \lim_{n \to \infty} \frac{1}{n} \sum a_i$.
A: Well, CLT doesn't tell sum is Normal. It tells the normalized average $(\bar X_n-n\mu)/\sqrt{n}\,\sigma$ has an asymptotic distribution that is normal. The sum may be Gamma or not depending on the parameters.
A: Assuming the rate (or scale) parameters are all the same, then you are correct:

*

*the sum of independent gamma distributed random variables has a gamma distribution, with shape parameter equal to the sum of the shape parameters of the individual distributions

But gamma distributions with a large shape parameter are close to normal distributions with the same mean and variance, and (after adjusting for mean and variance) get closer as the shape parameter increases.  Here is a case where the rate is $1$ and the shape, mean and variance are all $100$: the gamma density is in black and the normal density in red

