Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $(X_n)_{n\geq 1}$ be a sequence of real valued random variables defined on $\Omega$. Suppose that there exists a sequence $a_n \underset{n\rightarrow \infty}{\longrightarrow} 0$ of positive reals such that $(X_n)_{n\geq 1}$ is bounded in probability: \begin{equation} X_n = O_{\mathbb{P}}(a_n) \end{equation} i.e. $\forall$ $\epsilon>0$, $\exists$ $K_{\epsilon}>0$, $\mathbb{P}(|X_n/a_n|>K_{\epsilon})<\epsilon$ for all $n\in \mathbb{N}$.
Since $a_n \underset{n\rightarrow \infty}{\longrightarrow} 0$, $X_n\overset{\mathbb{P}}{\longrightarrow} 0$. Without further assumptions, is there a standard way to numerically estimate the rates of convergence of the sequence $(X_n)_{n\geq 1}$ when one can generate an arbitrary number of trajectories?