It is well known that a random variable being Gamma distributed with integer shape parameter $k$ is equivalent to the sum of the squares of $k$ normally distributed random variables.
But what can I say about a gamma distributed random variable with non-integer $k$? Is there any other interpretation other than the Gamma distribution at all?
If $X\sim{\cal G}(\alpha,1)$ and $Y\sim{\cal G}(\beta,1)$ are independent then$$X+Y\sim{\cal G}(\alpha+\beta,1)$$ In particular, if $X\sim{\cal G}(\alpha,1)$, it is distributed with the same distribution as $$X_1+\cdots+X_n\sim{\cal G}(\alpha,1)\qquad X_i\stackrel{\text{iid}}{\sim}{\cal G}(\alpha/n,1)$$for any $n\in\mathbb N$. (This property is called infinite divisibility.) This means that, if $X\sim{\cal G}(\alpha,1)$ when $\alpha$ is not an integer, $X$ has the same distribution as $Y+Z$ with $Z$ independent from $Y$ and $$Y\sim{\cal G}(\lfloor\alpha\rfloor,1)\qquad Z\sim{\cal G}(\alpha-\lfloor\alpha\rfloor,1)$$It also implies that integer valued shapes $\alpha$ have no particular meaning for Gammas.
Conversely, if $X\sim{\cal G}(\alpha,1)$ with $\alpha<1$, it has the same distribution as $YU^{1/\alpha}$ when $Y$ is independent from $U\sim{\cal U}(0,1)$ and $$Y\sim{\cal G}(\alpha+1,1)$$And hence the distribution ${\cal G}(\alpha,1)$ is invariant in $$X \sim (X'+\xi)U^{1/\alpha}\qquad X,X'\sim{\cal G}(\alpha,1)\quad U\sim{\cal U}(0,1)\quad \xi\sim{\cal E}(1)$$
