Recurrence for dependent random walks.

Question

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 ?

Answer 1

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.

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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 $\mu=0$.