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 [math]\displaystyle{ X_n }[/math]: [math]\displaystyle{ X_n = Z_1 + \dots + Z_n }[/math] where [math]\displaystyle{ Z_1, Z_2, \dots, Z_n }[/math] are independent m-dimensional vectors with a given multivariate distribution,

then if [math]\displaystyle{ m = 1 }[/math], [math]\displaystyle{ E(|Z_i|) \lt \infty }[/math] and [math]\displaystyle{ E(Z_i) = 0 }[/math], or if [math]\displaystyle{ m = 2 }[/math] [math]\displaystyle{ E(|Z^2_i|) \lt \infty }[/math] and [math]\displaystyle{ E(Z_i) = 0 }[/math],

the following holds: [math]\displaystyle{ \forall \varepsilon \gt 0, \Pr(\forall n_0 \ge 0, \, \exists n\ge n_0, \, |X_n| \lt \varepsilon ) = 1 }[/math]

However, for [math]\displaystyle{ m \ge 3 }[/math], [math]\displaystyle{ \forall A\gt 0, \Pr(\exists n_0 \ge 0, \, \forall n\ge n_0, \, |X_n| \ge A) = 1. }[/math]

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