# Applying Central Limit Theorem

I'm practicing probability and came across this question which I think could be solved by appealing to the Central Limit Theorem.

Question:

Let $$X_i, i\geq 1$$ be i.i.d. uniform random variables on $$[0,1]$$. Let $$S_n=X_1 + \dots + X_n$$. Find constants $$\alpha$$ and $$\beta$$ such that the limit $$\lim_{n\to\infty} \mathbb{P}(S_n<\alpha n + n^\beta)$$ exists and is strictly between $$\frac{1}{2}$$ and $$1$$.

My attempt:

I tried re-arranging to get it in a form where I could use CLT, using $$\alpha=\beta=\frac{1}{2}$$ to get

$$\mathbb{P}(S_n<\alpha n + n^\beta) = \mathbb{P}(\frac{S_n - n\cdot \frac{1}{2}}{\frac{1}{\sqrt{12}} \cdot \sqrt{n}}<\sqrt{12})$$

And then taking limits via CLT to get $$\Phi(\sqrt{12})$$.

Is this right? Is there a simpler way? Also if anyone could provide insight into why this question is being asked/is interesting I'd appreciate that.

The CLT yields that $$\sqrt{12n}\left(\frac{S_n}{n} - \frac{1}{2}\right) \to N(0, 1),$$ so we have that $$\lim_{n \to \infty} P\left(\sqrt{12n}\left(\frac{S_n}{n} - \frac{1}{2}\right)< z\right) = \Phi(z).$$ Rearranging, $$\lim_{n \to \infty} P\left(S_n < z\sqrt{\frac{n}{12}}+\frac{n}{2}\right) = \Phi(z),$$ so if you take $$z = \sqrt{12}$$ we get that $$1/2 < \lim_{n \to \infty} P(S_n < \sqrt{n} + n/2) = \Phi(\sqrt{12}) < 1$$. So the choice $$\alpha = \beta = 1/2$$ is a correct choice.
But moreover, it is the only actual choice. We have that $$P\left(S_n < \alpha n + n^\beta\right) = P\left(\sqrt{12n}\left(\frac{S_n}{n} - \frac{1}{2}\right) < \sqrt{12n}\left(\frac{\alpha n + n^\beta}{n} - \frac{1}{2}\right)\right) = P\left(\sqrt{12n}\left(\frac{S_n}{n} - \frac{1}{2}\right) < \sqrt{12n}\left(\alpha - \frac{1}{2}\right) + \sqrt{12}n^{\beta-1/2}\right).$$
Since the LHS of the the inequality goes to a $$N(0, 1)$$ random variable, if we want the probability to be in $$[1/2, 1]$$, we need the RHS to be nonnegative and finite as $$n \to \infty$$. But then clearly we need $$\alpha = \beta = 1/2$$.
The interpretation is that we are trying to match how $$S_n$$ "grows". Certainly we expect $$S_n$$ to on average increase by $$1/2$$ every increment of $$n$$ as $$n \to \infty$$ (that is, $$P(S_n > n/2) = 1/2$$ and $$P(S_n > \alpha n)$$ for $$\alpha \neq 1/2$$ is either 0 or 1); the issue is capturing how the distribution changes as $$n \to \infty$$ so, and the term $$n^{1/2}$$ captures the fact that the CLT tells you that $$S_n$$ will have second moment that is asymptotically on the order of $$n$$ (since $$S_n/n$$ has second moment that vanishes on the order of $$n$$).