Does random walk have more concentration surrounding the origin?

Question

Consider a simple random walk $S_n$ on one dimension, starting at $0$. In this case, $S_n$ fluctuates between $-\infty$ and $\infty$, but intuition says that it might stay more often in an interval surrounding the origin. To formulate this, consider an interval $A=[-d, d]$, and introduce a random variable $1_A(S_n)=1$ if $S_n\in A$ and $0$ otherwise.

Consider for some $f\in [0,1]$, the probability

$$Pr[\lim_{n\rightarrow \infty} \frac{\sum_{j=0}^n 1_A(S_j)}{n}\geq f]$$

I am not quite sure whether the definition is sensible, as the limit might not exist, but in that case, one could replace by $\lim\sup$ or $\lim\inf$.

The question is, what's the property of this probability?

Note that for finite ergodic Markov chains, similar problems can be answered easily by looking at stationary distributions, but for here, essentially we have a null-recurrent Markov chain.

Answer 1

This is an answer to the reformulation of the problem in comments. Using Markov's inequality for $x>0$, we obtain \begin{align*} \mathbb P\left(\frac{\sum_{j=0}^n I_A(S_j)}{n}>x\right) &\le \mathbb E\frac{\sum_{j=0}^n I_A(S_j)}{nx} = \frac{1}{nx}\sum_{j=0}^n \mathbb P(S_j\in A )\le \frac{1}{nx}\sum_{j=0}^n \frac{C}{\sqrt {j+1}}\\ &\le C_1\frac{\sqrt n}{nx}\to 0, \quad n\to \infty. \end{align*} Here, I used an estimate for concentration function $\mathbb P(S_j\in [-d,d])\le C(2d+1)/\sqrt{j}$ for all $j$ and a constant $C$ that does not depend on $j$ and $d$, see Theorem 9 in Chapter 3 of Petrov's book "Sums of independent random variables".

Clearly, $\sum_{j=0}^n I_A(S_j)$ is of order $\sqrt n$.