Chung–Fuchs theorem explained

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} :$$X\_n\; =\; Z\_1\; +\; \backslash dots\; +\; Z\_n$$where Z _{1,} Z _{2,} ... , Z _{n} are independent m-dimensional vectors with a given multivariate distribution,

then if

m = 1 , E(|Z _{i|)} < inf ty and E(Z _{i)} = 0 , or if m = 2 E(|Z ^{2} _{i|)} < inf ty and E(Z _{i)} = 0 ,

the following holds:

However, for

m \ge 3 ,$$\backslash forall\; A>0,\; \backslash Pr(\backslash exists\; n\_0\; \backslash ge\; 0,\; \backslash ,\; \backslash forall\; n\backslash ge\; n\_0,\; \backslash ,\; |X\_n|\; \backslash ge\; A)\; =\; 1.$$

References

• .
• "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