Higher order conserved charge fluctuations inside the mixed phase

General formulas are presented for higher order cumulants of the conserved charge statistical fluctuations inside the mixed phase. As a particular example, the van der Waals model in the grand canonical ensemble is used. The higher order measures of the conserved charge fluctuations up to the hyperkurtosis are calculated in a vicinity of the critical point (CP). The analysis includes both the mixed phase region and the pure phases on the phase diagram. It is shown that even-order fluctuation measures, e.g., scaled variance, kurtosis, and hyperkurtosis, have only positive values in the mixed phase and go to infinity at the CP. For odd-order measures, such as skewness and hyperskewness, the regions of positive and negative values are found near the left and right binodals, respectively. The obtained results are discussed in a context of the event-by-event fluctuation measurements in heavy-ion collisions