I need to prove that for an aperiodic, irreducible Markov Chain $X_n$ with stationary distribution $\pi$ holds that $P_x[X_n=j]\to\pi(j)$ exponentially fast. I found some proof of that statement but using reversibility of $\pi$ which I do not want to assume. Does some one knows a good reference for that?
Alternatively, using the coupling proof of the convergence statement is there a way of saying that the stopping time of the coupling has an exponential distribution?
Sketch: use irreducibility to conclude that there is only one stationary distribution. Use aperiodicity to conclude that there are no other eigenvalues of modulus $1$ other than $1$. Use Gerschgorin's theorem to conclude that there no eigenvalues of modulus larger than $1$ (maybe this is obvious).
Now if the transition probability matrix were diagonalizable, then you would be done. If not, then largest possible entry (other than the $1$ corresponding to the stationary distribution) in the Jordan form of $P^n$ is $n^{|S|-1} \lambda^n$ where $|S|$ is the number of states and $\lambda$ is the next largest eigenvalue away from $1$. Asymptotically this decays faster than, say, $\left ( \frac{|\lambda|+1}{2} \right )^n$, which is exponentially fast.