Poisson process at infinity

Question

Is the following probability well-defined?

Let $(X_t)_{t>0}$ be Poisson process ~ (pt)

$$\lim_{N\to\infty}P(X_{N/(2p)}< N ) = ?$$

How do I deal with the fact that the time is infinite ($t=\frac{N}{2p}$) and the number of arrivals if infinite (N)?

Answer 1

I'm not quite sure what you're asking, but here's my best guess. If $X_t$ is a Poisson process with intensity $p$, you seem to be asking about the limit as $N \to \infty$ of ${\mathbb P}(X_{N/(2p)} < N)$. Or maybe it's asking about something like $\mathbb P(\lim_{N \to \infty} X_{N/(2p)}/N < 1)$.

Hint: $X_{N/(2p)}$ is the sum of $N$ independent Poisson random variables $X_{1/(2p)}$, $X_{2/(2p)} - X_{1/(2p)}$, ..., $X_{N/(2p)} - X_{(N-1)/(2p)}$, each with expected value $1/2$. Use the Laws of Large Numbers. The Weak Law addresses the first question, and the Strong Law the second.