If I have a set of independent random variables, $X_1,X_2,...,X_n$ which each have mean 0 and variance $\frac{1}{n}$, how would I use the cauchy schwarz inequality to come up with the bound
$$ \sum^n_{i=1}E(|X_i|)\leq \sqrt{n} $$
I have tried splitting the absolute value into a product and then using cauchy schwarz but I cant seem to get this bound.
On
since we know that
$$\frac{1}{n} = var[X] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = \mathbb{E}[X^2]$$
you can use Jensen's inequality
$$(\mathbb{E}[|X_i|])^2 \leq \mathbb{E}[X_i^2] = \frac{1}{n} \implies \mathbb{E}[|X_i|] \leq \frac{1}{\sqrt{n}}$$
from which we have that
$$\sum_{i=1}^n \mathbb{E}[|X_i|] \leq \frac{n}{\sqrt{n}} = \sqrt{n}$$
$E[|X_i| |X_i|] \geq E[|X_i|]^2$ by Jensen's inequality, therefore you get, $$\sum E(|X_i|) \leq \frac{1}{\sqrt{n}} n = \sqrt{n}$$ Note that, cauchy-schwarz can be proved by Jensen's inequality, but I'm not sure of the reverse.