I have a 2-state Markov chain with the following transition matrix
${\displaystyle P={\begin{bmatrix}1-p&p\\1&0\end{bmatrix}} }$, where $0 < p < 1$.
Initially, we are in State 1.
Let $X_n$ be the number of times State 2 is visited by time $n$.
I would like to have some concentration inequality for $X_n$ deviating from it's expected value.
That is something like this:
$$ P(|X_n - \mathbb{E}[X_n]| > \mathbb{E}[X_n]) < f\left(\frac{1}{n}\right), $$
where $f$ is, say, some polynomial.
Does such an inequality hold? If yes, what would be a function $f$?