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.

notwhat you want to consider, because $\dfrac{\sum_{j=0}^n 1_A(S_j)}{n+1}$, which is the fraction of time $\le n$ the process spends in the interval $A$, goes to $0$ a.s. as $n \to \infty$. $\endgroup$ – Robert Israel Jul 7 '16 at 0:59