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 $X_{n}$:

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

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

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

the following holds:

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

However, for $m\geq 3$,

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

References

• Cox, Miller (1963), The theory of stochastic processes, London: Chapman and Hall Ltd.
• "On the distribution of values of sums of random variables" Chung, K.L. and Fuchs, W.H.J. Mem. Amer. Math. Soc. 1951 no.6, 12pp