Typical dimensions of large neutronic systems are often two orders of magnitude greater than the mean free path of the neutrons. Such high dominance ratio systems represent a particularly challenging issue when performing Monte Carlo criticality simulations. As a matter of fact, these simulations are contaminated by a cycle-to-cycle correlation that strongly slows down the flux convergence. In this paper, we will first discuss the link between the dominance ratio and the cycle-to-cycle correlations that are responsible for the poor flux convergence. Then, we will present a new and original technique to assess the dominance ratio of a given Monte Carlo simulation. It consists of fitting the relaxation process of the neutron field after an initial excitation from a fission source with a Dirac delta function shape. Having showed that these flux convergence issues are dominance ratio driven, we will then propose the use of an "independent replicas" approach to deal with the underprediction bias in statistics. The different theoretical points presented in this paper will be verified on a pin cell test case simulated with the Monte Carlo code TRIPOLI4. Additional results based on a three-dimensional pressurized water reactor core calculation are provided to confirm the reliability of the fitting technique described.