Consider two probabilistic algorithms, say $A(\_)$ and $B$. Algorithm $A$ takes as input a positive integer and algorithm $B$ takes no input. Both compute the expected value of the "same" discrete random variable $X$ as follows:
$A(n)$ simulates the behavior of a system comprising $n$ components and computes the expected value of $X$.
$B$ simulates the behavior of a system comprising an unbound (i.e., infinte) number of components and computes the expected value of $X$.
Note that the value of $X$ computed by $A(\_)$ tends to be more precise as $n$ grows. Moreover, according to many simulations, the values computed by $A(n)$ tend to get closer to that of $B$ a $n$ grows. We have the hypothesis that, from a topological point of view, $B$ is somewhat the limit of $(A(n))$ for computing $X$. The goal is to prove this hypothesis.
My background is on formal verification of concurrent systems (without probabilities). There, we have some notions of equivalence that are well known (e.g., bisimilarity, invariance). However, having probabilistic choice, renders such techniques useless (as far as I am aware of).
My question is twofold:
First, does it make sense to think of $B$ as the limit of $(A(n))$? If so, are there techniques to prove it? If not, what am I missing?
Second, the situation above may be simplified if thinking about two probabilistic algorithms $C(n)$ and $D(n)$, with a positive $n$ integer as parameter, which behave alike for large values of $n$. How to prove (or disprove) that both have the same limit or are "equivalent" for large values of $n$?
Any answer, thought, and correction will be greatly appreciated, as well as directions to relevant literature.
Thanks!