Tuesday, September 29, 2009
02:30 PM in HAAS 111
Jason Swanson
Department of Mathematics, University of Central Florida

Fluctuations of the empirical quantiles of independent Brownian motions

Abstract

We consider n independent, identically distributed one-dimensional Brownian motions, Bj(t), where Bj(0)has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by Bj:n(t), and we are interested in a sequence of quantiles Qn (t) = Bj(n):n , (t) where j(n)|n → ∈ (0,1). This sequence converges in probability in C [0,∞) to q(t), the α-quantile of the law of Bj (t). Our main result establishes the convergence in law in C [0,∞) of the fluctuation processes Fn = n1/2 (Qn - 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 B1/4, the fractional Brownian motion with Hurst parameter H = 1/4. For example, it is a quartic variation process, it has Hölder continuous paths with any exponent γ > 1/4, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of B1/4.