On the rate of convergence for countable Markov chains

Let $Z_n$ he an ergodic Markov chain with state space $\Omega=\{0,1,\dots\}$ and $\tau_{ij}=\min\{n\ge 1\colon Z_n=j\ (Z_0=i)\}$. We find necessary and sufficient conditions for $\mathbf M\tau_{ij}^{\gamma}<\infty$ ($\gamma\ge 1$). It is proved that the condition $\mathbf M\tau_{ij}^{\gamma}<\infty$ is sufficient for the existence of $C(k)<\infty$ such that
$$ |p_{ij}^{(n)}-\pi_j|\le C(k)n^{1-\gamma}\mathbf M\tau_{ik}^{\gamma},\qquad n=1,2,\dots, $$
where $p_{ij}^{(n)}=\mathbf P\{Z_n=j\mid Z_0=i\}$, $\displaystyle\pi_j=\lim_{n\to\infty}p_{ij}^{(n)}$.