Exact bounds on the closeness between the Student and standard normal distributions

TL;DR

This paper derives nearly optimal bounds on how close the Student distribution with p degrees of freedom is to the standard normal distribution, with small errors even for moderate p, and extends these bounds to compare different Student distributions.

Contribution

It provides the first precise, nearly optimal bounds on the Kolmogorov distance between Student and normal distributions, including bounds between different Student distributions.

Findings

Bounds are nearly optimal and small for moderate p.

The bounds apply to both Student-normal and Student-Student comparisons.

The results improve understanding of distributional closeness in statistical theory.

Abstract

Upper bounds on the Kolmogorov distance (and, equivalently in this case, on the total variation distance) between the Student distribution with p degrees of freedom (SD_p) and the standard normal distribution are obtained. These bounds are in a certain sense best possible, and the corresponding relative errors are small even for moderate values of p. The same bounds hold on the closeness between SD_p and SD_q with q>p.