Scaling limits for the Lego discrepancy

TL;DR

This paper develops a perturbative method to compute the distribution of the Lego discrepancy, a chi-squared statistic, for large numbers of bins and data points, and explores its asymptotic behavior through a phase diagram.

Contribution

It introduces a novel perturbative approach to analyze the distribution of the Lego discrepancy in large-sample regimes and provides a phase diagram for its limiting behavior.

Findings

Calculated the moment generating function perturbatively for large M and N.

Derived a phase diagram describing the distribution limits as M and N grow.

Provided insights into the asymptotic behavior of the Lego discrepancy.

Abstract

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.