Using strong law of large numbers to prove transience

Question

I'm trying to work my way through a problem which defines $N_t$ as a Poisson process of rate $\lambda$ and $ X_n = N_n − n,\quad\text{for }\; n = 0, 1, 2, \ldots $

I've explained why $X_n$ is a Markov chain and I've found the transition probabilities to be

$$ P(X_{n+1} = k+i \mid X_n = k) = e^{-\lambda}\frac{\lambda^{i+1}}{(i+1)!} $$ for $i \geq0$, $$ P(X_{n+1} = k-i \mid X_n = k) = e^{-\lambda} $$ for $i = 1$, and $0$ otherwise.

Now I'm supposed to use the strong law of large numbers to show that the train is transient if $\lambda \neq 1$. The strong law relies on each random variable in the sum being i.i.d, and I really have no idea how to approach this.

Any help you could offer would be much appreciated.

Answer 1

Hint:

1. Clearly, $$X_n = N_n-n = \sum_{j=1}^n \underbrace{(N_j-N_{j-1}-1)}_{=:\xi_j}$$ Since the Poisson process $(N_t)_{t \geq 0}$ has independent and stationary increments, the random variables $\xi_j$, $j \geq 1$, are independent and identically distributed. Show that $$\mathbb{E}(\xi_1) = \lambda-1.$$
2. By step 1 and the strong law of large numbers, we get $$\lim_{n \to \infty} \frac{X_n}{n} = \mathbb{E}(\xi_1)=\lambda-1 \quad \text{a.s.}$$
3. Deduce that $\lim_{n \to \infty} X_n = \infty$ if $\lambda>1$, and $\lim_{n \to \infty} X_n = -\infty$ almost surely if $\lambda<1$.