# Gamma distribution to normal approximation

I know that if there are i.i.d. random variables $X_i \sim N(0,\sigma^2)$ for $i = 1, 2, \cdots, N$, then $Y = X_1^2 + X_2^2 + \cdots + X_N^2$ follows $\Gamma(N/2,2\sigma^2)$.

When $N$ is sufficiently large number, Gamma distribution can be approximated as normal distribution $N(\mu_n,\sigma_n^2)$. I can find closed form of $\mu_n$ and $\sigma_n^2$ experimentally, but I can not derive the closed form of them mathematically. And also I can not define the sufficient large value $N$.

My question is how to find $\mu_n$ and $\sigma_n^2$ mathematically and how can I define the lower bound of $N$ for approximating gamma to normal distribution? (many references say that proper degree of freedom is 30)

• Are you assuming that the $X_i$ are independent? In that case $Y \sim N(0,N\sigma^2)$ and $Y$ does not have a $\Gamma$ distribution. Or do you want $Y = X_1^2 + \dots + X_N^2$? Commented Jan 16, 2018 at 5:33
• @HansEngler Sorry, There is some mistakes. Y = $X_1^2 + \cdots X_N^2$ is right. and I assume that $X_i$ are independent
– kyub
Commented Jan 16, 2018 at 5:37
• $Y$ is $\chi^2$ and not gamma? Commented Jan 16, 2018 at 5:47
• @AnyAD I know that chi square and gamma are related.When $X_i \sim N(0,1)$, $Y$ is $\chi^2$.
– kyub
Commented Jan 16, 2018 at 5:55
• A sequence of gamma random variables with shape parameters tending to infinity will converge to normal with matching mean and variance. It's a limit theorem. Proof is by MGFs. (I don't know what you mean by sufficiently large N. Sufficiently large for what purpose? Do you have a particular application in mind.) Commented Jan 16, 2018 at 23:54

By the central limit theorem, $\sqrt{N}\left(\frac{1}{N}(X_1^2 + \cdots + X_N^2) - 1\right)$ converges in distribution to $N(0, 2)$, since $X_1^2$ has mean $1$ and variance $2$. You can manipulate this to find your $\mu_N$ and $\sigma^2_N$. Regarding how large $N$ should be for a "good" approximation, you would need something like the Berry-Esseen theorem to give a quantitative statement.
• Thanks for the reminder (+1) of Berry-Esseen, which I have not thought about for a while, and the Wikipedia page with more recent info as to the value of $C$ than I remember seeing before. Commented Jan 16, 2018 at 23:50
Comment continued. The proof that $Y \sim \mathsf{Gamma}(shape=N/2, scale=2\sigma^2)$ can be done using moment generating functions.
Graphically, you can get an idea of the approach of $\mathsf{Gamma}(n, 1/\sqrt{n})$ [solid blue density curve] to $\mathsf{Norm}(\sqrt{n}, 1)$ [black broken curve], where $n = 4, 16, 64, 100.$ @angryavian has already mentioned Berry-Esseen as an approximate quantitative approach.