# Using strong law of large numbers to prove transience

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.

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 $$\lambda<1$$.