Concentration inequality for Lipschitz Function

Question

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(X_n:\Omega\rightarrow \mathbb{R}^m)_n$ be a sequence of i.i.d. random variables and let $L:\mathbb{R}^m\rightarrow [0,\infty)$ be Lipschitz. Let $\mu_n:=\frac1{n} \sum_{k=1}^n \delta_{X_k}$. Are there conditions under which: $$ \mathbb{P}\left(|\mathbb{E}_{X\sim\mu_n}[L(X)]-\mathbb{E}_{X\sim Law(X_1)}[L(X)]|\geq t\right)\leq \exp\left( -t^2 \right), $$ where $c>0$ is some constant?