Monte Carlo methods typically require us to construct very large vectors of numbers. In doing so it is often of great importance that the generated random numbers are independent.
My question here, as someone who knows next to nothing about random number generators, is: Is there a risk that the shortcomings of some, or all, random number generators influence the end result of the Monte Carlo simulation in a noticeable way with some bias or subtle dependence between the random numbers?
I heard someone say that the commonly used Mersenne twister could only guarantee independence of the elements in up to 623 elements long vectors, which is way smaller than the typical length of a Monte Carlo sample. Don't know if I misunderstood that, but it would be nice if someone could shed some light on the matter.