Let $X_1, ..., X_n$ be i.i.d. samples drawn from a pdf $f(x)$ on the real line. The kernel density estimator is defined as follows,
$$\hat{f_n}(x) = \frac{1}{nh}\sum_1^n K(\frac{x-X_k}{h})$$
where $K:\mathbb{R}\to [0, \infty)$ satisfying $\int_{-\infty}^{\infty}K(x) = 1$ and $h>0$.
How to prove that $\Vert \hat{f_n}(x) - f \Vert_1$ is sub-Gaussian w.r.t. $\frac{1}{\sqrt{n}}$ i.e.
$$\mathbb{P}(|\Vert \hat{f_n}(x) - f \Vert_1 - \mathbb{E}(\Vert \hat{f_n}(x) - f \Vert_1)|\geq t) \leq 2e^{-\frac{nt^2}{2}}$$
where
$$\Vert \hat{f_n}(x) - f \Vert_1:=\int_{-\infty}^{\infty}|\hat{f_n}(x)-f(x)|dx$$
