Scaling limits for the Lego discrepancy

For the Lego discrepancy with M bins, which is equivalent with a chi^2-statistic with M bins, we present a procedure to calculate the moment generating function of the probability distribution perturbatively if M and N, the number of uniformly and randomly distributed data points, become large. Furthermore, we present a phase diagram for various limits of the probability distribution in terms of the standardized variable if M and N become infinite.