Let $K_t$ be a Poisson process with rate $1$ and $X_n=K_n-n$ $, \ \ \ n\in \mathbb{N}$
Then we know that $X_n$ is a recurrent Markov chain. I am asked to determine whether the chain is null or positive recurrent.
I tried to compute the mean return time to zero, but, using the law of total probability, I got huge sums...
If $T=\min(n \geq 1 : X_n=0 |X_0 = 0) $ then I get:
$$ \mathbb{P}(T=k) = \sum_{t_1\not = 1}\sum _{t2\not = 2} \cdots \sum_{t_{k-1}\not = k-1} \mathbb{P}(K_1 = t_1)\mathbb{P}(K_2 = t_2)\cdots \mathbb{P}(K_{k-1}=t_{k-1})\mathbb{P}(N_k=k) $$
It does not look right....
How can I go on?
thank you very much