Question about Markov chain derived from a Poisson process

Question

Let $(N_t)$ be a Poisson process of rate $\lambda$. Define $$ X_n = N_n − n,\quad\text{for }\; n = 0, 1, 2, \ldots $$ Explain why $(X_n)$ is a Markov chain and give its transition probabilities. Using Stirling’s formula or otherwise, show that the chain is recurrent if and only if $\lambda = 1$. If $\lambda = 1$, is it null recurrent or positive recurrent?

I think the state space is all of the integers and the chain is irreducible. I think the transition probabilities are $$ P(X_{n+1} = k+r \mid X_n = k) = e^{-\lambda}\frac{\lambda^{r+1}}{(r+1)!} $$ for an integer $k$ and a non-negative integer $r$, and $$ P(X_{n+1} = k-r \mid X_n = k) = e^{-\lambda} $$ for $r = 1$ and $0$ otherwise.

I am having trouble with proving that the chain is recurrent, I think I should use the fact that a chain is recurrent if and only if $$ \sum_{n=0}^\infty p_{ii}^{(n)}=\infty $$ I am not sure how to do this, I tried finding the stationary distribution but this is very difficult.

Please help.

Answer 1

OKay have I have done this. I am not going to give you the solution, but I will offer you some hints:

You attempted approach to think about $p_{ii}^{(n)}$ is correct

$p_{ii}^{(0)}=1$ - agreed?

what is $p_{ii}^n$? well I think this is the probability of Poisson process jumping $n$ steps in time interval $n$. well the number of jumps in a time interval length $n$ is distributed as ....? so this probability is .... ?

The term should be a nice form $\bigg(\dfrac{n}{e}\bigg)^n/n!$ stirling's formula tell you $n!$ is approximately what? so the term behaves like $\dfrac{1}{n^\alpha}$ for $\alpha = ? <1$ therefore the series diverges.

Answer 2

I think we can try to solve the recurrence relation. I don't know if this will work, but here is an idea:

Let $t_i$ be expected amount of time to arrive at $0$ given you are currently at state $i$,.

$t_1 = p_{10} \times 1 + p_{11} (t_1+1) + p_{12} (t_2 +1) + p_{13} (t_3+1) + ... $

where $p_{ij}$ can be worked out and note $p_{ij} = p_{0,j-i}$ whenever $j\geq i$

get a system of these questions and see you can solve these. (though I hope there is an easier way) I will get back to you later. I need to do something for the next hours, sorry!