Approximating the distribution of the $L_q$-norm of a random point in a $d$-dimensional cube

Abstract

In this note, we assess the accuracy of CLT-based approximations for the volume of intersection of the $d$-dimensional cube $[-1,1]^d$ and an $L_q$-ball centred at the origin; this is clearly equivalent to approximating the distribution of the $L_q$-norm of a random point in a $d$-dimensional cube centered at 0. The approximations are CLT-based where to improve the normal approximation we use the first term in the Edgeworth expansion. We have included a section analysing the information obtained from ChatGPT in response to prompts regarding this theory; in our case, ChatGPT answers were not very helpful. Illustrations of the approximation formulae, as the radius of the ball increases, for different values of $d$ and $q$ are also given, alongside lines showing a Monte Carlo simulation of the intersection volume.