It is known that every infinitely divisible random variable is the limit in law of a sequence of compound Poisson random variables (see for instance Theorem 1.2.18 of Lévy Processes and Stochastic Calculus).

This can be shown as follows. Let $X$ be a infinitely divisible random variable with characteristic function $\Phi_X(t) = \mathbb{E}[\mathrm{e}^{\mathrm{i} t X}]$. Since $X$ is infinitely divisible, $\Phi^{1/n}$ is a valid characteristic function. Hence, for every $n \in \mathbb{N}$, we can define $X_n$ the compound Poisson random variable* with characteristic function $\exp\left(n\left(\Phi_X^{1/n}(t)-1 \right)\right)$. Moreover, we have $\Phi_{X_n}(t) \rightarrow \Phi_X(t)$ for all $t\in\mathbb{R}$, therefore $X_n \rightarrow X$ in law.

*Reminder: a compound Poisson random variable has a characteristic function of the form $\exp\left(\lambda\left(\Phi(t)-1\right)\right)$ where $\Phi$ is a characteristic function itself and $\lambda >0$.

Can we quantify the rate at which the convergence of $X_n$ to $X$ holds?

I am thinking, for instance, of something in the spirit of the Berry-Esseen theorem, bounding $\sup_x \lvert F_n(x) - F(x) \rvert $ with $F$ (resp. $F_n$) the cdf of $X$ (resp. $X_n$).

More generally, I would be interested by any result that quantifies the rate of convergence in law.