Weak Hölder Convergence of Smoothed Uniform Quantile Process

Abstract

It is well known that the paths of the Brownian bridge B are (with probability one) of Hölder regularity α for any α < 1/2 and the uniform quantile process converges weakly to the Brownian bridge in the Skorokhod space D[0,1]. We prove that this weak convergence holds too in Hölder spaces H _α [0,1]. The polygonal smoothing of the quantile process (instead of the quantile function, we use its interpolate at points (k/n,U _k:n ) where U _k:n is the kth order statistic of a uniform sample (U ₁,…, U _n )) gives us this convergence for any order α < 1/2. The convolution smoothing brings rather to an order α depending on the choice of window of the convolution kernel in this convergence.

Keywords:

• Brownian bridge
• Convolution smoothing
• Hölder space
• Polygonal smoothing
• Quantile process

Mathematics Subject Classification:

• 60B10
• 60F05
• 62G30