Average of Monte Carlo simulations of continuous functions again continuous?

Question

I hope the following question is clear:

Suppose, we have a continuous functions $f:\mathbb{N}^2 \rightarrow \mathbb{N}$. Now, suppose we run Monte Carlo simulations on the function, where the input parameters are distributed according to a continuous probability distribution. Suppose we calculate the sample mean of the MC-simulations outcomes.

Would it make sense to define continuity on the entire process? What would continuity of this process actually mean? Would it make sense to define continuity on the result after infinite many MC-simulations?

Thank you in advance.

Answer 1

I don't understand your question that well. From what I understand it is akin to population distribution and sampling distribution for the process. If one is continuous, the other is continuous too. To further describe, let us say you have normal distribution with $N(\mu,\sigma^2)$ where $\mu$ and $\sigma^2$ are true parameters, then sampling parameters $(\bar x, \bar S)$ are the for the sampling distribution. Similarly, average of monte carlo distribution from a true probability distribution is the same as the sampling distribution with random bias (owing to pseudo random simulations) if that makes you comfortable.