Wiener Process, Donsker’s Theorem, and Brownian Bridge

In this chapter, we get to know the Wiener process, which is the starting point for many other stochastic processes. This process is accompanied by the Wiener measure on the \(\sigma \) -field of Borel sets on the function space \({\mathrm {C}} := {\mathrm {C}}[0,1]\) . According to the title of this book, a limit theorem must not be missing, and that is Donsker’s theorem, which represents a far-reaching generalization of the of Lindeberg–Lévy central limit theorem. With the help of the Wiener process, the Brownian bridge emerges, which plays an important role in nonparametric statistics.