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?
Best Answer
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.