I have n random variables $X_0^1,...X_0^n$ which are i.i.d. with law $\mu_0$ such that \begin{align*} \exists\;\epsilon>0:\; \mu_0(\lambda,\infty)=\mathcal{O}(\exp\{-\epsilon\lambda\})\quad\text{as}\quad\lambda\rightarrow\infty. \end{align*}
i.e. the $X_0^{i}$ have sub-gaussian initial law. Let $\nu_0:=\frac{1}{n}\sum\delta_{X^{i}_0}$. You can assume the density $V_0$ of $\nu_0$ to exist in $L^2$. I'm trying to get an $L^2$, i.e. \begin{align*} \Vert V_0\Vert_2^2\leq 1. \end{align*}