obtain exponential inequality using symmetric random variables and rademacher random variables I'm new to mathematical stats and came here to get some help for my homework problem in advanced statistics.
The problem is:
Let $X_1, \cdots, X_n$ be independent, not necessarily identical random variables. Assume that each $X_i$ is symmetric, meaning that $X_i  \overset{d}{\equiv} -X_i$. Prove that $P \left[ \dfrac{\sum X_i}{\sqrt{\sum X^2_i}} \geq t \right] \leq e^{-t^2/2}, \forall t>0$.
The problem gives hint that we could use rademacher random variable $\epsilon$ using the fact that $X \overset{d}{\equiv}\epsilon X$. There are no distributional assumptions on random variables $X_i$. 
Can anyone give me a hint?
 A: The symmetrization hint can be used in the following way: if $\varepsilon_1, \ldots, \varepsilon_n$ are i.i.d. Rademacher random variables (independent of $X_1, \ldots, X_n$), we have
$$
P\left(\sum_{i=1}^n X_i\ge t\|X\|_2\right)=P\left(\sum_{i=1}^n \varepsilon_i X_i \ge t\| X\|_2\right).
$$
Now we can think about $X_1, \ldots, X_n$ being fixed vectors, and look only at the randomness over $\varepsilon_1, \ldots, \varepsilon_n$. For example, using the formalism of conditional expectations,
$$
P\left(\sum_{i=1}^n \varepsilon_i X_i \ge t\| X\|_2\right)=\mathbb E\left[P\left(\sum_{i=1}^n \varepsilon_i X_i \ge t\| X\|_2\,\middle|\, X\right)\right].
$$
Now to bound the inner term, we can apply Hoeffding's inequality to $\sum_i Y_i$, where each $Y_i=\varepsilon_i X_i$ is bounded in absolute value by $|X_i|$ (recall that now, all the $X_i$'s are fixed). Hence,
$$
P\left(\sum_{i=1}^n \varepsilon_i X_i \ge t\| X\|_2\,\middle|\, X\right)\le \exp(-t^2/2),
$$
and we can take expectations on both sides.
