Relation To Other Stochastic Processes
If W(t) is a standard Wiener process (i.e., for t ≥ 0, W(t) is normally distributed with expected value 0 and variance t, and the increments are stationary and independent), then
is a Brownian bridge for t ∈ .
Conversely, if B(t) is a Brownian bridge and Z is a standard normal random variable, then the process
is a Wiener process for t ∈ . More generally, a Wiener process W(t) for t ∈ can be decomposed into
Another representation of the Brownian bridge based on the Brownian motion is, for t ∈
Conversely, for t ∈
The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as
where 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.
Read more about this topic: Brownian Bridge
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