Brownian bridge of the Brownian bridge based on the Brownian motion is, for "t" ∈ [0, T] Conversely, for "t" ∈ [0, ∞] The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as where formula_9 are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem). A Brownian bridge is the result of Donsker's theorem in the area of empirical processes. It is also used in the Kolmogorov–Smirnov test in the area of statistical inference. A standard Wiener process satisfies "W"(0) = 0 and is therefore "tied down" to the origin, but other points are not
Brownian excursion Brownian excursion process, formula_1, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. Another representation of a Brownian excursion formula_1 in terms of a Brownian motion process "W" (due to Paul Lévy and noted by Kiyoshi Itō and Henry P. McKean, Jr.) is in terms of the last time formula_3 that "W" hits zero before time 1 and the first time formula_4 that Brownian motion formula_5 hits zero after time 1: Let formula_7 be the time that a