Brownian motion is a central object of probability theory. Roughly speaking, we could perform a space-time rescaling of a symmetric nearest neighbor random walk on the integer lattice such that the limiting process has normally distributed increments and continuous paths.
We provide different principles of construction: Firstly, via the Kolmogorov–Chentsov theorem on continuous modifications and secondly via a Fourier series with random coefficients in the fashion of Paley–Wiener or of Lévy. Finally, we show the functional central limit theorem (invariance principle) which states that the properly rescaled partial sum process of centered square integrable i.i.d. random variables converges in path space to Brownian motion.
KeywordsBrownian Motion Continuous Path Path Space Path Property Reflection Principle
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