Divergence of sample quantiles

TL;DR

This paper investigates the convergence behavior of sample quantiles, showing they tend to the true quantile when left and right quantiles coincide, but diverge almost surely otherwise, with detailed asymptotic properties.

Contribution

It generalizes known results by characterizing divergence of sample quantiles and their limsup and liminf in cases of non-convergence.

Findings

Sample quantiles converge when left and right quantiles are equal.

Sample quantiles diverge almost surely when they differ.

Limsup and liminf of diverging quantiles correspond to right and left quantiles.

Abstract

We show that the left (right) sample quantile tends to the left (right) distribution quantile at p in [0,1], if the left and right quantiles are identical at p. We show that the sample quantiles diverge almost surely otherwise. The latter can be considered as a generalization of the well-known result that the sum of a random sample of a fair coin with 1 denoting heads and -1 denoting tails is 0 infinitely often. In the case that the sample quantiles do not converge we show that the limsup is the right quantile and the liminf is the left quantile.