I have a question about how to prove concentration bounds of random walks landing on points. Here is a simplified model:
Consider a simple random walk on $[-10,-9,...,-1,0,1,...,9,10]$ that starts from point 0, and moves +1 or -1 with probability 1/2, except there is only one choice +1 for the left endpoint -10 and -1 for the right endpoint 10. Then the stationary distribution has a probability 1/20 at point 0.
Can we prove that for a $n$-step random walk, the number of times that it lands on 0 is concentrated around $n/20$? Namely, the probability of landing on 0 with more than $n/10$ times is exponential small, i.e., $$\Pr[\textit{the random walk lands on 0 more than } n/10 \textit{ times}] \le C^{-n} \qquad \textit{for some }C>1?$$
Any suggestion on reference or key words will be helpful. For example, although I know the Chernoff bound is a standard tool to show an exponential small tail, I am not aware of any variant of it that can be applied here.
Yes, such bounds are by now quite well known, see e.g.
[1] https://projecteuclid.org/journals/annals-of-applied-probability/volume-8/issue-3/Chernoff-type-bound-for-finite-Markov-chains/10.1214/aoap/1028903453.full