Chung–Fuchs theorem

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.

Specifically, if a position of the particle is described by the vector

n

{\displaystyle X_{n}}

:

n

=

n

1

2

, … ,

are independent m-dimensional vectors with a given multivariate distribution, then if

, the following holds:

∀ ε > 0 , Pr ( ∀

< ε ) = 1

> 0 , Pr ( ∃