Fluctuations of the empirical quantiles of independent Brownian motions

TL;DR

This paper studies the fluctuations of empirical quantiles of independent Brownian motions, showing they converge to a Gaussian process with properties similar to fractional Brownian motion with Hurst parameter 1/4.

Contribution

It establishes the convergence in law of the scaled empirical quantile fluctuations to a Gaussian process and provides an explicit covariance formula, revealing local properties akin to fractional Brownian motion.

Findings

Fluctuation processes converge to a Gaussian process.

The limit process has H"older continuity with exponent less than 1/4.

The process exhibits quartic variation and negative correlation of increments.

Abstract

We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles $Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This sequence converges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$-quantile of the law of $B_j(t)$. Our main result establishes the convergence in law in $C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limit process $F$ is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that $F$ has many of the same local properties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H = 1/4$. For example, it is a quartic variation process, it has H\"older continuous paths with any exponent $\gamma < 1/4$, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of $B^{1/4}$.