Let $\{X_i\}_{i\in\mathbb{N}}$ be a sequence of random variables taking values in $\{\pm e_1,\pm e_2\}$, where $\{e_1,e_2\}$ is the standard basis of $\mathbb{R}^2$. If $\{X_i\}$ are i.i.d. uniformly distributed over $\{\pm e_1,\pm e_2\}$, then the simple random walk $S_N$ on $\mathbb{Z}^2$ have the mean square displacement given by $\mathbb{E}[\|S_N\|^2] =N$. We also know that the random walk $S_n$ is recurrent, i.e., $$ \sum_{n=1}^{\infty}\mathbb{P}(S_{2n}=0)=+\infty. $$ Question Suppose now that the random variables $X_i$ $(i=1,2,\ldots)$ have some kind of dependence between its coordinates (but the steps $X_{i}'s$ are independent) so that the mean square displacement now obeys the following inequality for any $N\in\mathbb{N}$ $$ C_1N^2 \leq \mathbb{E}[\|S_N\|^2] \leq C_2N^2, $$ where $0<C_1<1/2$ and $1/2<C_2<1$ are positive constants. Is this inequality enough to assures that this random walk is transient ?

  • $\begingroup$ By "same kind of dependence", do you mean "some kind of dependence"? $\endgroup$ – user940 Oct 11 '13 at 3:46
  • $\begingroup$ Sorry Byron, should be "some" isntead of same. $\endgroup$ – Leandro Oct 11 '13 at 19:30
  • $\begingroup$ I have been assuming that the $X$ random vectors are independent, though individually the coordinates of $X$ may be correlated. Is this what you mean by "some dependence"? $\endgroup$ – user940 Oct 12 '13 at 23:41
  • $\begingroup$ I changed the question. Because the first version of it was not clear and it seems that I need to be more precise about what kind of dependence I want to consider. Thanks again. $\endgroup$ – Leandro Oct 13 '13 at 1:53
  • $\begingroup$ Your new question still says that the steps are independent. $\endgroup$ – user940 Oct 13 '13 at 1:58

Let $\mu=\mathbb{E}(X)$ and $\sigma^2=\mathbb{E}\|X-\mu\|^2$. Taking expectations in $$\|S_N\|^2\leq 2\|S_N-N\mu\|^2+2\|N\mu\|^2,\tag1 $$ we get $$\mathbb{E}\|S_N\|^2\leq 2N\sigma^2 +2N^2 \|\mu\|^2.\tag2 $$ If the left hand side of (2) goes to infinity faster than $N$, then $\mu\neq0$ and the walk is necessarily transient.

Added: The strong law of large numbers gives ${S_N\over N}\to\mu$. If the random walk were recurrent, $S_N$ would visit the state $0$ infinitely often, and the only possible limit point of ${S_N\over N}$ would be $0$. So recurrence forces $\mu=0$.

  • $\begingroup$ Dear Byron, thank you for the answer. One more question: to conclude that this random walk is transient did you use some Ergodic theorem (which I guess in my case do not holds) or SLLN ( which probably holds) ? $\endgroup$ – Leandro Oct 11 '13 at 20:18
  • $\begingroup$ @Leandro It follows from the strong law of large numbers which definitely holds in your problem. $\endgroup$ – user940 Oct 12 '13 at 23:29
  • $\begingroup$ Byron, thanks a lot for the comments and answer. You absolutely right if the steps $\{X_1,X_2,\ldots\}$ are independent. Unfortunatelly I poorly stated my problem. What I really wanted is to assume that the steps are dependent, for example, each coordinate is a Ising spin random variable interacting via an infinite range potential. In fact, I should make more precise the dependence, but I thought that this speed $cN^2$ it was enough to conclude transience independently of the type of the dependence. $\endgroup$ – Leandro Oct 13 '13 at 1:32

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