# Convergence speed of discrete approximation

Here I asked the question about approximating the function $$g(x) := \mathbb{E}(f(x,Y))$$, where $$x \in R$$ and $$Y$$ is a random variable. If you follow the link you will see that $$g(x)$$ can be approximated by finite sum $$\frac{1}{n}\sum_{i=1}^nf(x,\mu_i)$$ with any precision we desire.

My question: whats is the speed of the convergence? Can we deduce how large $$n$$ should be to make error $$\leq \epsilon$$

Feel free to impose any reasonable conditions on $$f$$ and distribution of $$Y$$

• Are you looking for something on the lines on concentration inequalities? – sudeep5221 Jul 6 at 13:38
• @sudeep5221, I am looking for anything that could tell me how good my discrete approximation is – Markoff Chainz Jul 6 at 14:31