$\textbf{Problem:}$ let {$X_{n} : n ≥ 0$} be any symmetric random walk on $\mathbb{Z^{2}}$ whose jumps have finite second moment. That is, $X_{0} = 0$ , {$X_{n} − X_{n−1} : n ≥ 1$} are mutually independent, identically distributed, symmetric ($X_{1}$ has the same distribution as $−X_{1}$), $\mathbb{Z^{2}}$-valued random variables with finite second moment. Show that {$X_{n} : n ≥ 0$} is recurrent in the sense that $\mathbb{P}(\exists n\ge 1 : X_n =0)=1$.
here my thoughts, first is (my trouble) we don't have nearest neighbor random walk. So , I don't know how to define the time of first return to $0$ (origin) (here i mean I am not sure whether returning time to origin is finite or infinite ). If I can define that , i think we can find the value of $\mathbb{P}$(first time to return $0<\infty$) and I am thinking if i can find the probability of first return to origin then I thought I can find the probability of $n^{th}$ return (because of independency). However it didn't work out that way because {$X_{n} − X_{n−1} : n ≥ 1$} are mutually independent. So the problem is how to use mutually independency ? Besides, since $X_{n}$ are symmetric , clearly we get $\mathbb{E}$[$|X_{n}|$] is finite . But how can I find the value ?
I couldn't see the whole picture. I also couldn't find anything about non-nearest neighbor random walks in $\mathbb{Z}^d$. Could you please help me to solve this problem ? Thanks in advance.