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I am solving $I = \int_{0}^{1}f(x)dx$ by Monte Carlo, e.g $I = E[f(U)]$ where $U\sim unif([0,1])$ so have

$I = E[f(U)]\approx \hat I_M\colon= \frac{1}{M}\sum_{m=1}^Mf(u_m)$

I am interested in how quickly $\hat I_M$ converges to $I$, i.e. how quickly $|\hat I_M - I|\rightarrow 0$ as $M\rightarrow \infty$. I do not understand in which convergence sense I am looking at? Is it in simply in 'absolute convergence sense'?

Thanks in advance for any help!

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www.stanford.edu/class/cme308/OldWebsite/notes/chap3.pdf (section 3.6 and especially section 3.8 for the central limit theorem). –  Learner Apr 27 '13 at 12:21
    
Awesome thanks! That has really helped –  Kane Blackburn Apr 27 '13 at 14:02
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