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 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 ${\displaystyle X_{n}}$:

$${\displaystyle X_{n}=Z_{1}+\dots +Z_{n}}$$

where ${\displaystyle Z_{1},Z_{2},\dots ,Z_{n}}$ are independent m-dimensional vectors with a given multivariate distribution,

then if ${\displaystyle m=1}$, ${\displaystyle E(|Z_{i}|)<\infty }$ and ${\displaystyle E(Z_{i})=0}$, or if ${\displaystyle m=2}$ ${\displaystyle E(|Z_{i}^{2}|)<\infty }$ and ${\displaystyle E(Z_{i})=0}$,

the following holds:

$${\displaystyle \forall \varepsilon >0,\Pr(\forall n_{0}\geq 0,\,\exists n\geq n_{0},\,|X_{n}|<\varepsilon )=1}$$

However, for ${\displaystyle m\geq 3}$,

$${\displaystyle \forall A>0,\Pr(\exists n_{0}\geq 0,\,\forall n\geq n_{0},\,|X_{n}|\geq A)=1.}$$