# Approximation and Monte Carlo simulation.

I am a bit up over my head here, I will present an argument and then I hope you guys will say if my reasoning is correct or what should be changed, ultimately I am hoping to say something qualified about the size of $\varepsilon$ as to be defined.

Let $X$ be some quite unmanageable random variable (it is an integral of a stochastic process) and assume of which I would like to approximate $E[X]$. Using $M$ operations (approximating it with trapezoidal discrete integral on M points) I can get $Y_M$ such that $X-Y_M$ is asymptotically $O(1/M^2)$. I can get $N$ of such $Y_M$'s (by simulation), i.e. $Y_M^1,\dots Y_M ^N$ and approximate $E[X]\approx \tilde{S}_N^M = 1/N \sum_{i=1}^{N} Y_M ^i$ by the central limit theorem the error from doing this is asymptotically $O(1/\sqrt{N})$. So if I choose $M= \sqrt[3]{N}$ my error will be asymptotically $O(1/\sqrt{N})$ and if I consider the error $$\varepsilon =\frac{\lvert E[X] -\tilde{S}_N^M \rvert}{E[X]}$$ it will be in the order of magnitude $1/\sqrt{N}$. In particular if I get a value of say $0.05$ and have $N=10^6$ I can expect that $\varepsilon \approx 0.1 \%$ or that I can be certain of my result up to $1\%$?

• When you do the simulation, you can also estimate the error of the simulation as $(\sum Y^2_i/N- (\sum Y_i/2)^2)/(N-1)$. This will give you the Monte-Carlo estimator variance. – Yulia V Oct 9 '14 at 16:49

There are two possible problems with your approach.

The first is theoretical, but will have no practical consequences if your stochastic process is modeling some physics process: For a general random distribution having a finite expectation value, there is no guarantee that the variance is finite; and if it is not, then the conditions of the central limit theorem are not met.

The second is practical and important: If you integrate your stochastic process (run your simulation) you need to run long enough that the influences of the initial conditions becomes small ("start-up" or "warm-up" time). Now if you are doing a million sample points, you don't really want to have to decorrelate from starting conditions a million times. You would rather run a sequential Mnte Carlo, taking points far enough apart in simulation step time that they are effectively decorrelated. And here is the rub: Many physical processes when simulated by Monte Carlo will have very long decorrelation times.

A perfect example is finding the expectation value of any physical quantity using lattice quantum chromodynamics with dynamic quark fields. The decorrelation times are manageable for unphysically large quark masses, but if you let the quark masses get small (as the physical masses of the up and down quarks really are) the correlation time grow as something like the square of the inverse quark mass. So you have to be careful, when quoting uncertainties, that you have studied step correlations properly.

• Hi Mark! Thanks for your reply, it would be very possible that I have not considered the problems. In my case though I know where the process is supposed to start (this can actually be estimated at a previous step of my work) and I know that the unmanageable stochastic variable has at least mean and variance. I should have written that I realise now. Do you know if the error estimation is correct then though? :) – Henrik Oct 9 '14 at 17:10
• Yes it is, up to a multiplicative constant that depends (linearly, I think) opn the variance. – Mark Fischler Oct 10 '14 at 0:20