As is well known, a symmetric random walk on $\mathbb{Z}^d$ (the lattice of $d$ dimensional vectors with integer components) is recurrent if and only if $d=1,2$. In particular it is transient for $d=3$.

What about a random walk on $\mathbb{Z}^2$, where the probability to move right is equal to the probability to move left, and the probability to move forward is equal to the probability to move backward, but they are not necessary all $0.25$? (They do sum to $1$ of course). Well it turns out that this is recurrent as well.

My question is this: Is there a non-computational proof of this? i.e. can one show in some rigorous sense that this is "the same" as a truly symmetric $2$-$d$ random walk and thus recurrent?

edit: perhaps I should replace "non computational" with "slick" or "short". I was given a problem on an exam that required as a lemma the fact that these walks are recurrent. When I went over the official solution, I was surprised that the lecturer stated without proof that in this type of walk the probability of returning to the origin at time n is of order 1/n, as though this was some immediate corollary of the symmetric case (where this holds). I guess he was just being sloppy...

  • $\begingroup$ One can make independent the displacements of each coordinate (which are, as you note in a comment below, very much dependent in the original model) if one passes in continuous time, using Poisson processes and splitting them. Then the explanation of the recurrence of your models becomes that the integral of $1/\sqrt{t}\cdot1/\sqrt{t}=1/t$ diverges at infinity. Is this the kind of thing you are after? $\endgroup$
    – Did
    Commented Oct 4, 2013 at 16:06
  • $\begingroup$ this book, amazon.com/Random-Electric-Networks-Mathematical-Monographs/dp/… is available free on line and is nice. It deals with graph processes where you weight the edges & pick one at weighted random.In yr case vertical have weight 1 & horizontal 3, or vice versa. $\endgroup$
    – mike
    Commented Oct 4, 2013 at 20:14

1 Answer 1


In fact, there is a theorem in Durrett's book "probability theory and examples"(P186):

When $d=2$, for all iid random variables $X_n, n=1,2,\dots$ if $S_n/n^\frac{1}{2}\Rightarrow$ a normal distribution, then $S_n$ is recurrent

Since the 2d random walk is symmetric, by the central limit theorem you can see that $S_n/n^\frac{1}{2}\Rightarrow$ a normal distribution, then $S_n$ is recurrent


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