# Let X be$Gamma(\alpha, \lambda)$Prove $(\lambda X - \alpha)/\sqrt{\alpha} \xrightarrow{d} N(0,1)$ as $\alpha \rightarrow \infty$ and $\lambda$is fixed [closed]

First of all the continuity lemma is stated as follows:

Let $$\mu_n, n=1,2, \dots$$ be a sequence of distributions, and $$\varphi$$ the associated characteristic function.

1. If $$\mu_n \xrightarrow{w} \mu$$, then for all $$t \in \mathbb{R}$$ $$\varphi_n(t) \rightarrow \varphi(t)$$, where $$\varphi$$ is the characteristic function of $$\mu$$.

2. If for all $$t \in \mathbb{R}$$ $$\varphi(t) := lim_{n \rightarrow \infty} \varphi_n(t)$$ exists, and is continous at $$t=0$$, then $$\varphi$$ is the characteristic function of a distribution $$\mu$$, and $$\mu_n \xrightarrow{w} \mu$$

From here the question goes as follows:

Let X be $$Gamma(\alpha, \lambda)$$ distributed (notice that $$\alpha$$ might be a non-integer). Prove, via characteristic functions and the Continuity Lemma, that $$$$\frac{\lambda X - \alpha}{\sqrt{\alpha}} \xrightarrow{d} Normal(0,1)$$$$ as $$\alpha \rightarrow \infty$$ and $$\lambda$$ is fixed

## closed as off-topic by Lee David Chung Lin, YuiTo Cheng, Cesareo, marwalix, Adrian KeisterMay 8 at 13:29

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Lee David Chung Lin, YuiTo Cheng, Cesareo, marwalix, Adrian Keister
If this question can be reworded to fit the rules in the help center, please edit the question.

Basically what you should do is find the MGF of $$\frac{\lambda X -\alpha}{\sqrt{\alpha}}$$ and then take the limit as $$\alpha \to \infty$$ and check that it is equal to $$\exp{(t^2/2)}$$

(I say MGF, but use CF)