Weak convergence of the scaled median of independent Brownian motions

Abstract

We consider the median of n independent Brownian motions, denoted by M n (t), and show that √ n M n converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be Hölder continuous with exponent γ for all γ < 1/4.

Bahadur–Kiefer representations for time dependent quantile processes

“…Corollary 1. For any 0 < ρ < 1/2, T > 1 and δ > 0 we have Observe that (20) combined with the compact LIL pointed out in (13), immediately imply that…”

“…For a random particle motivation to look at such problems consult the Introduction in [13], where possible fractional Brownian motion generalizations are hinted at.…”

“…Swanson [13] using classical weak convergence theory proved that an appropriately scaled median of n independent Brownian motions converges weakly to a mean zero Gaussian process. More recently Kuelbs and Zinn [9], [10] have obtained central limit theorems for a time dependent quantile process based on n independent copies of a wide variety of random processes, which may be zero or perturbed to be not zero with probability 1 [w.p.1] at zero.…”

Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

“…(v) Gaussian process in a paper by Swanson. This process was introduced in [25], and arises as the limit of normalized empirical quantiles of a system of independent Brownian motions. The covariance is given by…”

Central limit theorem for functionals of a generalized self-similar Gaussian process

“…(a) bifractional Brownian motion (see [8]), (b) subfractional Brownian motion (see [2]), (c) an 'arcsine' Gaussian process introduced in a paper by Jason Swanson [20], and (d) two self-similar Gaussian processes that form the decomposition of a process discussed in a paper by Durieu and Wang [6].…”