mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Chung–Fuchs theorem, named after Chung Kai-lai and

Wolfgang Heinrich Johannes Fuchs Wolfgang Heinrich Johannes Fuchs (May 19, 1915, Munich – February 24, 1997) was a British mathematician specializing in complex analysis. His main area of research was Nevanlinna theory. Fuchs received his Ph.D. in 1941 from the University of C ...

, states that for a particle undergoing a

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

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^2_i, ) < \infty

and

E(Z_i) = 0

, the following holds:

\forall \varepsilon > 0, \Pr(\forall n_0 \ge 0, \, \exists n\ge n_0, \, , X_n,  < \varepsilon ) = 1

However, for

m \ge  3

\forall A>0, \Pr(\exists n_0 \ge 0, \, \forall n\ge n_0, \, , X_n,  \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 {{DEFAULTSORT:Chung-Fuchs theorem Eponymous theorems of physics