Chung–Fuchs theorem

In mathematics, the Chung–Fuchs theorem, named after Wolfgang Heinrich Johannes Fuchs and Chung Kai-lai, states that for a particle undergoing a 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 X n :

X n = Z 1 + . . . + Z n

where Z 1 , Z 2 , . . . , Z n are independent m-dimensional vectors with a given multivariate distribution,

then if m = 1 , E ( | Z i | ) < ∞ and E ( Z i ) = 0 , or if m = 2 E ( | Z i 2 | ) < ∞ and E ( Z i ) = 0 ,

the following holds:

∀ ϵ > 0 , Pr ( ∀ n 0 ≥ 0 , ∃ n ≥ n 0 , | X n | < ϵ ) = 1

However, for m ≥ 3 ,

∀ A > 0 , Pr ( ∃ n 0 ≥ 0 , ∀ n ≥ n 0 , | X n | ≥ A ) = 1 .