I've got problem with that task:
Consider $\{Z_n\}_{n>0}$ is iid with integer values with expected value $\mathbb EZ_1<0$ and $\{X_n\}_{n\ge0}$ is homogeneous Markov chain defined by $$ X_{n+1} = (X_n+Z_{n+1})_+$$$ n\ge1$, where $X_0$ is stochastic independent with $\{Z_n\}$. Using Pakes Lemma show, that irreducible $\{X_n\}$ is positive recurrent.
I found the Lemma: http://www.lsv.ens-cachan.fr/~monmege/teach/probas/TD2.pdf, p.2. I showed that $$p_{ij} = \begin{cases}P(Z_1\le-i), &\text{for } j=0\\P(Z_1=j-i), &\text{else}\end{cases}$$ And $$\mathbb E[X_{n+1} \mid X_n=i] < \infty$$But I dont know how to show or start with limit: $$\limsup_{i\to\infty}\mathbb E[X_{n+1}-X_n \mid X_n=i]<0$$
Firstly observe that \begin{align}\left(X_{n+1}-X_n \mid X_n=i\right)&=\left(\left( i+Z_{n+1}\right)_+-i\right)\\[0.2cm]&=\begin{cases}i+Z_{n+1}-i,& \text{if }Z_{n+1}+i\ge 0\\0-i,& \text{if }Z_{n+1}+i< 0\end{cases}\\[0.3cm]&=\begin{cases}Z_{n+1},& \text{if }Z_{n+1}\ge -i\\-i,& \text{if }Z_{n+1}<-i \end{cases}=\max{\{-i,Z_{n+1}\}}\end{align} Hence, $$\Bbb E[X_{n+1}-X_n \mid X_n=i]=\Bbb E\left[\max{\{-i, Z_{n+1}\}}\right]=\Bbb E\left[\max{\{-i, Z_1\}}\right]$$ where the last equality is true, since the $Z_n$'s are iid. Now, we will take the $\limsup$ of the this expression and we will interchange the order of $\limsup$ and $\Bbb E$ with the Reverse Fatou Lemma (RFL) to conclude:
\begin{align}\limsup_{i \to +\infty}\Bbb E\left[\max{\{-i, Z_1\}}\right]\overset{\text{(RFL)}}\le \Bbb E\left[\limsup_{i\to +\infty}\left(\max{\{-i, Z_1\}}\right)\right]&=\Bbb E[\max{\{-\infty,Z_1\}}]\\[0.2cm]&=\Bbb E[Z_1]<0\end{align}