Consider a $\mathbb{Z}$ valued markov chain $X_n$ which evolves as follows.
$$P(X_{n+1}=y | X_n) =\begin{cases} \frac{1}{2}, y=X_n+1, X_n-1, |X_n|>K \\ \frac{1}{4}, y = X_n-1 , y= X_n+1, |X_n|\leq K\\ \frac{1}{2}, y=X_n,|X_n|<K \end{cases}, $$
where $K>0$.
Basically, $X_n$ behaves like lazy simple random walk in $[-K,K]$ and simple random walk outside $[-K,K]$.
I am puzzled by the following inequality. $$P_0(X_n=y)\leq \frac{C}{\sqrt{n}}, \forall y$$ where $C$ is a constant independent of $y$ ,$P_0$ means the chain start at $0$.
I don't know how to approach this problem, the method for simple random walk which is basically binomial distribution does not work here.
Any hints or ideas?