I am curious as to what the sample mean distribution for a set of iid Gamma random variables is.
I know $\bar{X}$ has a Gamma distribution when $X_1, X_2 ... X_n$ are iid ~ exponential with mean $\theta$ and gamma distributed random variables $X_1 + X_2$ also follow a gamma distribution with parameters $(\alpha_1 + \alpha_2, \theta)$, for $X_1$ ~ $(\alpha_1, \theta)$ and $X_1$ ~ $(\alpha_1, \theta)$, but I can't find any information regarding the distribution of the sample mean for a set of gamma distributed random variables.
As a sidenote, when can we approximate $\bar{X}$ with a normal distribution? Ex: when can we say that the distribution of $\bar{X}$ for $X_1, X_2 ... X_n$ iid ~ exponential$(\theta)$ is approximately normal with mean $\theta$ and variance $\frac{\theta}{n}$, instead of saying it follows a gamma distribution?
Thank you for your answers in advance.
We can in fact relax the requirement that the sample be identically distributed: suppose for each $i = 1, 2, \ldots, n$, we have $$X_i \sim \operatorname{Gamma}(n_i, \theta),$$ where each $n_i$ is a positive integer and $\theta$ is a common scale parameter. Then $$S = \sum_{i=1}^n X_i \sim \operatorname{Gamma}\left(\sum_{i=1}^n n_i, \theta\right).$$ This is because each $X_i$ is the sum of $n_i$ iid exponential variables with mean $\theta$; e.g., $$X_i = Y_{i1} + Y_{i2} + \cdots + Y_{in_i},$$ thus the total sum is $$S = \sum_{i=1}^n \sum_{j=1}^{n_i} Y_{ij}.$$ In fact, we do not even require that the $n_i$ are positive integers; they may be arbitrary positive reals, and the result still holds. This can be shown by considering the moment-generating function of a gamma distribution. From here, the sample mean $\bar X = S/n$ is also gamma distributed, with shape $\sum n_i$ and scale $\theta/n$.