The Brownian Bridge

The Brownian bridge, or tied-down Brownian motion, is derived from the standard Brownian motion on [0, 1] started at zero by constraining it to return to zero at time t = 1. A precise definition is provided and its (Gaussian) distribution is computed. The Brownian bridge arises in a wide variety of contexts. An application is given to a derivation of the Kolmogorov–Smirnov statistic in non-parametric statistics in this chapter. An application to the Hurst statistic in special topics Chapter 27, to mention a few.