Show $Z_n= \frac{X_1 +\cdots + X_n}{\sqrt{n}}$ is uniformly sub-gaussian.

Question

As described above I would like to establish a tail bound for the $Z_n$, using that $X$ is bounded with zero mean and unit variance and $X_n$ are iid copies of $X$. So namely I would like to show that $P(\vert Z_n \vert \geq \lambda) \leq C\exp(-c\lambda^{2})$ for all $\lambda >0$ and $C,c>0$ independent of $n$ (uniformly sub-gaussian). I think that should be possible using some kind of Chernoff bound. I would really appreciate some Help.

Answer 1

• Let the $X_i$ lie in $[a,b]$ with probability $1$. Then Hoeffding's Lemma implies $E[e^{tX_1 / \sqrt{n}}] \le \exp(\frac{t^2(b-a)^2}{n})$.
• The Chernoff bound implies $P(Z_n \ge \lambda) \le \min_{t > 0} e^{-t\lambda} (E[e^{tX_1 / \sqrt{n}}])^n$. Plug in the above inequality and solve for $t$ to get a bound on the probability.
• $P(Z_n \le -\lambda)$ can be handled similarly.