Sorry, $Q$ can be obtained from $P$ by conditioning means that there exists a larger probability space $(\Omega^*, \mu)$ where the two measure can be "coupled" such that that $P$ is a marginal of $\mu$ and $Q$ is the conditional measure given by $\mu$ conditioning on certain event.

Thank you so much for the answer! I am interested in when $\mu\to 0$. So the constants can not implicitly depend on the step distribution $F$; they can certainly depend on $c_3$ since I can control this quantity while sending $\mu$ to zero. With a quick look over the paper, it seems that it doesn't use things like KMT coupling where the constants implicitly depend on $F$ that we have no control over. I will look more carefully at the papers, in case you can already spot a place where the constants might implicitly depend on $F$, I would really appreciate if you could point it out : )

@MateuszKwaśnicki Thanks again! In my case, the $X_i$ is an explicit (continuous) distribution supported on $(-\infty, \infty)$, but I will think about a condition so that this can be stated with generality.

@MateuszKwaśnicki thank you for the remark, I added the assumption that $\text{Var}(X_i) >\delta$.