Can I ask for advice on proceeding with this question? It is Problem 5.5.3 (page 282) of Samuel Karlin, Mark A. Pinsky's ''An Introduction to Stochastic Modelling'' 4th edition.
Let $\{X(A) : A \subseteq {R}^2 \}$ be a homogeneous Poisson point process in the plane, whose intensity is $\lambda$. Divided a square area of $(0, t] \times (0, t]$ into $n^2$ squares of length $d = \frac{t}{n}$. A reaction occurs if there are two or more points located within the same square of length $d$.
Objective: Determine the distribution of reactions in the limit as $t \to \infty$ and $d \to 0$ such that $td \to \mu > 0$.
My attempt: Let $X(t^2)$ denote the number of points in a square region of area $t^2$ then $X(t^2) \sim Poi(\lambda t^2)$ and $X(d^2) \sim Poi(\lambda d^2)$. The number of points in each square of length $d$ are i.i.d since these squares are disjoint. Consequently, the event of a reaction happening in each square of length $d$ is independent.
Denote $p = P(X(d^2) \geq 2) = 1 - e^{- \lambda d^2} - \lambda d^2e^{-\lambda d^2}$ then we are interested in the number of reactions in the $(0,t] \times (0,t]$. Denote $Y$ as the number of reactions in the $(0,t] \times (0,t]$, then \begin{equation} P(Y = k) = {n^2 \choose k}p^k(1-p)^{n^2 - k} = {n^2 \choose k}\left(1 - e^{- \lambda d^2} - \lambda d^2e^{-\lambda d^2}\right)^k \left(e^{- \lambda d^2} - \lambda d^2e^{-\lambda d^2} \right)^{n^2 - k} \end{equation}
What's strange is that I can't seem to force out the $td$ term. I'm guessing $Y$ will be a Poisson random variable in the limit of $t \to \infty$ and $d \to 0$.
Thanks.
Apply the law of large events, showing that $n^2 \left(1 - e^{- \lambda d^2} \left(1 + \lambda d^2 \right) \right) \to \lambda^2 \mu^2$ then we have $Y \sim Poi( \lambda^2 \mu^2 )$.