I want to propagate the uncertainty from a set of input variables $X$ with distribution $P_X$ through some function $Y=f(X)$, and obtain the output distribution $P_Y$. Doing this practically is quite simple, as we simply have to perform a Monte Carlo simulation for $M$ samples of $Y$, with the histogram of the results giving us what we want.
However, I am not quite sure how to define the objective of this simulation being the convergence of our sample frequency of $Y$ to a distribution, per se. I understand how to describe how Monte Carlo integration and such converges to singular values such as the mean via the law of large numbers as $\frac{1}{N}\sum_{i=1}^Nf(x_i)\rightarrow \mu_Y$. However, I don't know how to describe this same process for how the output converges to a desired distribution, as it does not seem to me like just describing the binning process of a histogram is the best way of conveying this goal.