Let $p>1/2$ and $(M_n)_{n\geq 0}$ be a Markov chain on $\mathbb{N}$ defined by the probability transitions:
with probability $p$, $M_{n+1}=\max(M_n-1,0)$
with probability $1-p$, $M_{n+1}=M_n+1$.
This is a simple birth and death process, and I know that it is a positive recurrent Markov chain with invariant measure $\mu(k)=\left(\frac{1-p}{p}\right)^k \left(1-\frac{1-p}{p}\right)$. My question is : what is known about the convergence of the $E_{M_0=0}(M_n)$ to $E_{X\sim \mu} X$? Is there any rate of convergence available?
More generally, consider $p_0>...>p_d$ such that $p_0+...+p_d=1$, and $X_n$ a process on $\mathbb{N}^d$ defined by the transition probabilities: with probability $p_i$, if $i>0$, $X_{n+1}(i)=X_n(i)+1$, and if $\{k>i|X_n(k)>0\}$ is not empty, letting $j=\min\{k>i|X_n(k)>0\}$, $X_{n+1}(j)=X_n(j)-1$. Note that in the case $d=1, p_0=p, p_1=1-p$, we get $X_n=M_n$.
I have found an invariant measure, namely $\mu(k_1,\dots,k_d)=\Pi_{i=1}^d\left(\frac{p_i}{p_0}\right)^{k_i} \left(1-\frac{p_i}{p_0}\right)$, so again what can be said on the convergence to the expectation, and do you have any references on this kind of generalization of the simple Birth and Death Process?
Thanks a lot.