The following is an exercise from my textbook.
Let $Y$ be a random variable with values in $\mathbb{N}_0$ and $Y_1, Y_2, \dots$ be independent copies of $Y$. Further let $X$ be a markov chain with $X_0$ independent from $Y_i$ and recursive defined by $$ X_{n+1} = \max{(X_n -1,0) + Y_{n+1}} $$
Interpretation: $Y_{n+1}$ new skier arrive at the ski-lift between the timepoints in which the $n$-th and $(n+1)$-th Person is being transported. Hence, $X_{n}$ would be the length of the queue after which $n$ persons are transported.
I was able to calculate the transition matrix but I'm stuck with the following (and clueless especially with the second one):
How can I show that $0$ is recurrent, if $\mathbb{E}{[Y]} \leq 1$ and that $0$ is transience, if $\mathbb{E}{[Y]} > 1$?