Let $A(t)$ be a unit rate poisson process. I want to show that $$\frac{A(t)}{t } \to 1 \quad \quad a.s. \quad as \ \ t \to \infty$$ I am able to show for all $n \in \mathbb N$ $$\frac{A(n)}{n } \to 1 \quad \quad a.s. \quad as \ \ n \to \infty$$
using the telescopic sum $$A(n)=\sum_{1 \leq i \leq n }A(i)-A(i-1)$$
How do I show for non integer t?
Let $S_n$ denote the time when the nth renewal occurs. Then we have $$ \frac{S_{A(t)}}{A(t)} \le \frac t{A(t) } \le \frac{S_{A(t) + 1}}{A(t)} $$ By the law of large number $$ \frac{S_{A(t)}}{A(t)} \to \frac1\lambda = 1 \quad \text{as}\ t \to \infty $$ since $A(t) \to \infty$ as $t\to\infty$. Similarly $$ \frac{S_{A(t) + 1}}{A(t)} = \frac{S_{A(t) + 1}}{A(t)+1}\frac{{A(t) + 1}}{A(t)} \to \frac1\lambda = 1 $$ Hence $$ \frac{A(t)}{t} \to \lambda = 1 $$