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Suppose that $X_ n sim text {Gamma} (nalpha , lambda)$ for all $n ge 1$, for fixed $alpha,lambda >0.$ Show that

$$frac {1} {sqrt n} left (X_n - frac {n alpha} {lambda} right ) implies Z ,$$

where $Z sim mathcal N (0,{sigma}^2)$ for some $sigma.$ Calculate $sigma.$

I have constructed a sequence random variables ${S_n }$ in the following way $:$

$S_1=X_1,S_2=X_2-X_1,S_3=X_3-X_2, cdots , S_n = X_n - X_{n-1} , cdots.$ Then I observed that $sumlimits_{k=1}^{n} S_k = X_n$ for all $n ge 1$. Also I observed that $Bbb E(S_n) = frac {alpha} {lambda}$ and $Bbb {Var} (S_n) = frac {alpha} {{lambda}^2}$ for all $n ge 1$. If moreover $S_n$'s can be shown to be independent then they are i.i.d. random variables with mean $frac {alpha} {lambda}$ and variance $frac {alpha} {{lambda}^2}$. Then by central limit theorem we can say that

$$frac {1} {sqrt n} left (X_n - frac {n alpha} {lambda} right ) implies Z$$ as $n rightarrow infty$ where $Z sim mathcal N (0,frac {alpha} {{lambda}^2})$. Hence ${sigma}^2 = frac {alpha} {{lambda}^2}$ i.e. $sigma =frac {sqrt alpha} {lambda}.$

But how do I prove that $S_i$'s are i.i.d. random variables keeping in mind the fact that $X_i$'s are independent random variables? Please help me in this regard.

Thank you very much.

This answer uses the the following fact.

If $X sim Gamma(alpha,lambda)$ and $Y sim Gamma(beta,lambda)$ are independent, then $X+Y sim Gamma(alpha+beta,lambda)$.

Hints: Let $(Y_i)_{i in mathbb{N}}$ be a sequence of independent identically distributed random variables such that $Y_i sim Gamma(alpha,lambda)$.

1. Show that $tilde{X}_n := sum_{i=1}^n Y_i$ satisfies $tilde{X}_n sim Gamma(n alpha,lambda)$.


2. Apply the central limit theorem to prove that $$frac{1}{sqrt{n}} left( tilde{X}_n - frac{n alpha}{lambda} right) stackrel{d}{to} Z$$ for $Z sim N(0,sigma^2)$ with $sigma^2 = text{var}(Y_1)$; here $stackrel{d}{to}$ denotes convergence in distribution.


3. Use the fact that $tilde{X}_n$ equals in distribution $X_n$ for each $n in mathbb{N}$ to conclude from Step 2 that $$frac{1}{sqrt{n}} left( X_n - frac{n alpha}{lambda} right) stackrel{d}{to} Z$$
   


4. Compute $sigma^2 = text{var}(Y_1)$ (...or look it up, e.g. on wikipedia).