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.
Best Answer
Hint: