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 {\displaystyle X_{n}} :
X n = Z 1 + ⋯ + Z n {\displaystyle X_{n}=Z_{1}+\dots +Z_{n}}
where Z 1 , Z 2 , … , Z n {\displaystyle Z_{1},Z_{2},\dots ,Z_{n}} are independent m-dimensional vectors with a given multivariate distribution, then if m = 1 {\displaystyle m=1} , E ( | Z i | ) < ∞ {\displaystyle E(|Z_{i}|)<\infty } and E ( Z i ) = 0 {\displaystyle E(Z_{i})=0} , or if m = 2 {\displaystyle m=2} E ( | Z i 2 | ) < ∞ {\displaystyle E(|Z_{i}^{2}|)<\infty } and E ( Z i ) = 0 {\displaystyle E(Z_{i})=0} , the following holds:
∀ ε > 0 , Pr ( ∀ n 0 ≥ 0 , ∃ n ≥ n 0 , | X n | < ε ) = 1 {\displaystyle \forall \varepsilon >0,\Pr(\forall n_{0}\geq 0,\,\exists n\geq n_{0},\,|X_{n}|<\varepsilon )=1}
However, for m ≥ 3 {\displaystyle m\geq 3} ,
∀ A > 0 , Pr ( ∃ n 0 ≥ 0 , ∀ n ≥ n 0 , | X n | ≥ A ) = 1. {\displaystyle \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