Let $X_i$ be i.i.d. random variables taking value from $\mathbb{Z}$ with mean $0$. Let $S_0=0$ and $S_n=X_1+...+X_n$. Let $H_n(x)=E\left[\sum_{i=0}^{n}I\{S_i=x\}\right]$ be expected number of visits to $x$ in the first $n$ steps. I wonder if $H_n(0)\ge H_n(x)$ for all $n$ and $x$. I don't know how to connect $H_n(0)$ and $H_n(x)$. I may think the first $j$ with $S_j=x$. Next, I wonder how to show that for each $\epsilon>0$, $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{|x|\le \epsilon n} H_n(x) = 1$. I have figured out that $\lim_{n\rightarrow\infty}\mathbb{P}\{|S_n|\le n\epsilon\}=1$ by L.L.N., but how to continue on? Finally, can I say this $S_n$ is a recurrent Markov chain?
I'd appreciate so much if anyone can help me on this problem.
Assume that $x \ne 0$. Let $T_x$ be the first hitting time to $x$. Then, by the Markov property, $$ E^0 \left[ \sum_{i=0}^{n} I\{S_i = x\} \right] = E^0 \left[ \sum_{i=T_x}^{n} I\{S_i = x\}, \ T_x \le n \right] = \sum_{k=1}^n P^0 (T_x = k) H_{n-k} (0).$$ Since $H_k (0)$ is non-decreasing with respect to $k$, $$ \sum_{k=1}^n P^0 (T_x = k) H_{n-k} (0) \le P^0 (T_x \le n) H_{n} (0) \le H_n (0).$$
We can construct transient one-dimensional random walks. Consider the case that $p(0,x) \asymp (\log|x|)^{-1}$ as $|x| \to \infty$ for example, where $p(,)$ denotes the transition probability.