Computable Bounds for Strong Approximations with Applications

TL;DR

This paper develops practical, computable bounds for the KMT inequality applicable to bounded i.i.d. variables, enabling improved change point detection and deviation analysis.

Contribution

It introduces a computable version of the KMT inequality with explicit constants and an empirical variant for unknown standard deviation, enhancing practical applications.

Findings

Derived a computable KMT inequality with explicit constants.

Developed an empirical inequality for unknown standard deviation.

Demonstrated applications in change point detection and hitting time probabilities.

Abstract

The Koml\'os$\unicode{x2013}$Major$\unicode{x2013}$Tusn\'ady (KMT) inequality for partial sums is one of the most celebrated results in probability theory. Yet its practical application has been hindered by a lack of practical constants. This paper addresses this limitation for bounded i.i.d. random variables. At the cost of an additional logarithmic factor, we propose a computable version of the KMT inequality that depends only on the variables' range and standard deviation. We also derive an empirical version of the inequality that achieves nominal coverage even when the standard deviation is unknown. We then demonstrate the practicality of our bounds through applications to online change point detection and first hitting time probabilities. As a byproduct of our analysis, we obtain a Cram\'er-type moderate deviation bound for normalized centered partial sums.