The Chung-Fuchs theorem, published by Kai-lai Chung and Wolfgang H. J. Fuchs in 1951, is a result in probability theory settling whether a random walk with zero mean returns infinitely often to any neighborhood of its starting point. The theorem shows that the answer depends on the dimension of the space the walk moves in: in one or two dimensions the walk is certain to return infinitely often to every neighborhood of the origin, while in three or more dimensions the walk eventually leaves for good and escapes to infinity. This sharp dependence on dimension is one of the clearest illustrations in probability theory of how random walk behavior changes qualitatively as the number of dimensions increases.
Facts
StatementFor a zero-mean random walk in m dimensions, the walk returns infinitely often to any neighborhood of the origin when m is 1 or 2, but escapes to infinity in three or more dimensions. 1 Sources
1. Chung-Fuchs theorem, Wikipedia
Lede section
In mathematics, the Chung-Fuchs theorem, named after Chung Kai-lai and Wolfgang Heinrich Johannes Fuchs, states that for a particle undergoing a zero-mean random walk in m-dimensions, it is certain to come back infinitely often to any neighborhood of the origin on a one-dimensional line (m = 1) or two-dimensional plane (m = 2), but in three or more dimensional spaces it will leave to infinity.
References section
Chung, K.L. and Fuchs, W.H.J. Mem. Amer. Math. Soc. 1951 no.6, 12pp
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.