# $X_1 ,\ldots, X_n$ are independent with distribution $X_i=\pm \sqrt{i}$ with probability $0.5$. Get the asymptotic distribution of the mean $\bar{X}$.

$X_1 ,\ldots, X_n$ are independent with distribution $X_i=\pm \sqrt{i}$ with probability $0.5$. Get the asymptotic distribution of the mean $\bar{X}$

Seems very easy. But I used mgf and find it's hard to calculate. To apply CLT I can make it to something like $$\sqrt{n}\frac{\sum_{i=1}^{n}X_i/\sqrt{n}}{n}$$But every term in the sum is related to $n$... Any ideas? Thanks!

• Hint: Can you show that $$\prod_{k=1}^n\cos\left(\frac{\sqrt{k}}{n}t\right)$$ converges when $n\to\infty$ and compute its limit? – Did Mar 20 '16 at 20:15
• sorry, I do not know how to figure out this limit... The only thing I know is the taylor expansion of the mgf of this question and your hint are alike.. – abc1m2x3c Mar 20 '16 at 22:25
• Sub-hint: When $x\to0$, $\cos(x)$ and $e^{-x^2/2}$ are similar. Could you compute the limit of the products above if each cosine was replaced by the corresponding exponential? – Did Mar 20 '16 at 22:38
• got it! Really appreciate your help! – abc1m2x3c Mar 21 '16 at 17:48