Let $N_t = N([0,t])$ denote a Poisson process with rate $\lambda = 1$ on the interval $[0,1]$.
I am wondering how I can use the Law of Large Numbers to formally argue that:
$\frac{N_t}{t} \rightarrow \lambda \quad \text{ a.s.} $
As it stands, I can almost prove the required result but I have to to assume that $t \in Z_+$. With this assumption, I can define Poisson random variables on intervals of size $1$ as follows
$N_i = N([i-1,i])$
where
$\mathbb{E}[N([i-1,i])] = \text{Var}[N([i-1,i])] = 1$
and
$N_t = N([0,t]) = \sum_{i=1}^t N([i-1,i]) = \sum_{i=1}^t N_i$
Accordingly, we can use the Law of Large Numbers to state the result above...
Given that $t \in \mathbb{R}_+$, this proof needs to be tweaked in some way... But I'm not exactly sure how to do it.
Intuitively speaking, I believe that the correct approach would be to decompose $N[0,t]$ into $N[0,\lfloor t\rfloor]$ and $N[\lfloor t\rfloor, t]$, and argue that the latter term $\rightarrow 0$ almost surely.
However, I'm not sure how to formally state this.