A remark on the asymptotic tightness in $\ell^{\infty}([a,b])$

TL;DR

This paper extends criteria for uniform tightness from continuous functions on (0,1) to bounded functions on [a,b], providing a broader understanding of asymptotic tightness in function spaces.

Contribution

It generalizes existing tightness criteria from continuous to bounded functions, enhancing theoretical tools for asymptotic analysis in function spaces.

Findings

Extended criteria for uniform tightness to $ ext{l}^{ ext{+ ext{infty}}}([a,b])$

Connected classical theorems to broader function classes

Provided a framework for asymptotic tightness in bounded functions

Abstract

In this note, we extend a simple criteria for uniform tightness in $C(0,1)$, the class of real continuous functions defined on $(0,1)$, given in Theorem 8.3 of Billingsley to the asymptotic tightness in $\ell^{+\infty}([a,b])$, the class of real bounded functions defined on $[a,b] $ with $a<b$, in the lines of Theorems 1.5.6 and 1.5.7 in van der vaart and Wellner.