Fluctuations of the empirical quantiles of independent Brownian motions

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 → α ∈ (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 = n (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. AMS subject classifications: Primary 60F05; secondary 60F17, 60G15, 60G17, 60G18, 60J65