The following question arose in one of my research projects. Before stating it, let me give a short background. We all know the **law of iterated logarithm**. It states that if $X_1,\ldots,X_n$ are i.i.d. mean $0$, variance $1$ random variables and if $S_n := X_1 + \ldots + X_n$, then almost surely,
$$\limsup_{n\rightarrow \infty} \frac{\pm S_n}{\sqrt{2n\log \log n}}=1~.$$
**My question is as follows:**

Is there a finite sample exponential concentration inequality for the quantity $\left|S_n/\sqrt{2n\log\log n}\right|?$ That is, suppose that $t > 1$ is fixed. Then can we bound the probability something like:$$\mathbb{P}\left(\left|\frac{S_n}{\sqrt{2n\log \log n}}\right|> t\right) \leq e^{-n^\alpha}$$ for some $\alpha > 0$?

Any help will be greatly appreciated.