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 $X_n$: $$X_n=Z_1+\dots +Z_n$$ where $Z_1,Z_2,\dots ,Z_n$ are independent m-dimensional vectors with a given multivariate distribution,

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

the following holds: $$\mathrm{\forall }\epsilon >0,\mathrm{Pr}(\mathrm{\forall }{n}_0\ge 0,\mathrm{\exists }n\ge n_0,|X_n|<\epsilon )=1$$

However, for $m\ge 3$, $$\mathrm{\forall }A>0,\mathrm{Pr}(\mathrm{\exists }{n}_0\ge 0,\mathrm{\forall }n\ge n_0,|X_n|\ge A)=1.$$

References

• Template:Citation.
• "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