Brownian bridge

The process is a generalization of the brownian bridge.

Alternatively, it is a Brownian bridge process conditioned to be positive.

The increments in a Brownian bridge are not independent.

A Brownian bridge is the result of Donsker's theorem in the area of empirical processes.

In this case, the process indexed by the indicator functions , for is in fact the standard brownian bridge B(x).

Just as a literal bridge is supported by pylons at both ends, a Brownian Bridge is required to satisfy conditions at both ends of the interval [0,1].

The Donsker theorem (1952) shows that converges in law to a Brownian bridge Komlós, Major and Tusnády established a sharp bound for the speed of this weak convergence.

Kolmogorov (1933) showed that when F is continuous, the supremum and supremum of absolute value, converges in distribution to the laws of the same functionals of the Brownian bridge B(t), see the Kolmogorov-Smirnov test.