Most papers I read related to Monte Carlo (MC) simulation method say that there is no rule on the minimum number of runs required. However, in book Risk Analysis by D. Vose there is the following formula for the estimation of the true mean:

\(n>\left ( \frac{\Phi ^{-1}\left ( \frac{1+\alpha }{2} \right )\sigma }{\delta } \right )^{2}\)

n - number of runs

Ф^(-1) - the inverse of the normal cumulative distribution function

sigma - standard deviation

alpha and delta - desired error and confidence

(the formula is derived based on the distribution of the estimate of the true mean (asymptotically) from the Central limit theorem and the assumption that If Monte Carlo sampling is used, each generated value Xi is an iid random variable).

The formula is derived for making random draws from a univariate distribution. My question is, would it be still valid if I have to make random draws using MC from a multivariate distribution? If not, is there any way to derive a similar formula for a multivariate case based on the same assumptions as for the univariate case?

Thank you.