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Expected loss of a single iteration of a Monte Carlo simulation, where the number of iterations varies?

This question is adapted from @Quantize’s answer to the following question: Monte Carlo simulation without replacement with n=10,000.

In particular, the question is:

Assume we have the following Monte Carlo simulation, with n (number of times we repeat the simulation) in a log scale:

What is the expected loss for a single iteration of the simulation, given the number of iterations is varied, as in the following diagram?

A:

If you want a simulation that is more faithful to the actual distribution of outcomes, then it is better to use the binomial than the geometric distribution. The binomial is the discrete approximation of the normal, and for any fixed $n$ the variance of the binomial is the same as that of the normal, but for fixed $\mu$ the variance of the binomial is smaller than that of the normal (i.e. the central limit theorem does not apply in the same way).

That being said, for the log-normal distribution, the standard deviation of the log-normal is $\sigma = \sqrt{2}$, and therefore, if you draw a random sample of size $n$ from that distribution, the probability that the log-loss exceeds some value $t$ is

$$ \mathbb P(Y > t) = 1 – \frac1{\sqrt{2\pi}}\int_t^\infty e^{ -x^2/2}\,dx = \frac{\Phi\left(-\frac{t}{\sqrt{2}}\right)}{\sqrt{2\pi}} $$

where $\Phi$ is the cdf of the standard normal.

This formula is not very good, however, because the cdf of

 

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