# Solved – Convergence from Gamma to Normal Distribution

I came across this problem:

Problem

If I have \$X_1, X_2, …, X_n\$ \$n\$ iid random variables which pdf is
\$\$
f_X(x) = begin{cases} dfrac{x^{mu-1} e^{-x}}{Gamma{(mu)}} &0<x<infty, \0&text{elsewhere}end{cases}
\$\$
namely a \$text{Gamma}(alpha=mu, beta=1)\$ distribution. Let \$nbar{X}_n= X_1+X_2+cdots+X_n\$. I need to prove that \$\$T_nstackrel{text{dist}}{stackrel{ntoinfty}{longrightarrow}}N(0,{1}/{mu})\$\$ where
\$\$T_n= frac{sqrt{n}(bar{X}_n-mu)}{bar{X}_n}\$\$

My Solution:

I thought of a possible solution in two steps:

• First, we need to find the pdf of \$bar{X}_n\$ and then of \$T_n\$.
• Then we take the limit of it and if we get a Normal distribution then, we solved the question.

begin{align*}
F_T(t) &= P(T le t)
\&
= P(frac{sqrt{n}(bar{X}_n-mu)}{bar{X}_n} le t)
\&
= P(frac{(bar{X}_n-mu)}{bar{X}} le frac{t}{sqrt{n}})
\&
= P(1- frac{mu}{bar{X}_n} le frac{t}{sqrt{n}})
\&
= P(frac{mu}{bar{X}_n} ge 1- frac{t}{sqrt{n}})
\&
= P(frac{bar{X}_n}{mu} le frac{sqrt{n}}{sqrt{n}-t})
\&
= P(bar{X}_n le frac{musqrt{n}}{sqrt{n}-t})
end{align*}

Now, I should do the integration \$int_0^frac{musqrt{n}}{sqrt{n}-t}\$ of the pdf of \$bar{X}_n\$. But it is not the same distribution as \$X_i\$. It is something else. This is where I stuck in my solution.

To clarify: My goal is to prove \$T_nstackrel{text{d}}{longrightarrow}Normal\$, not finding the distribution of \$bar{X}_n\$.

Any help will be appreciated!

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