Assume a Poisson point process in $[0,1]^2$, with intensity $\mu$. For any positive integer $n$, consider the division of the space into $4^n$ disjoint half open squares with side length $2^{-n}$. Let $N_n$ be the number of such half open squares that contain points of the process. Find the almost sure limit of $N_n$.

First, I note that for each such small square, it contains points with Poisson distribution of intensity $\mu4^{-n}$. Hence, the probability that each square contains at least a point is $1-e^{-\mu 4^{-n}}$. Hence, by independence of each squares, I can see that $N_n$ follows a binomial distribution $B(4^n,p_n)$, where $p_n=1-e^{-\mu 4^{-n}}$. I can see that the probability of any small square containing more than a point should go to $0$, so my guess is $N_n$ should converge to some Poisson distribution, but I don't have any idea on how to progress from here.

  • $\begingroup$ Hint: note that $p_n\sim\mu/4^n$. Now, what do random variables $S_k$ binomial $(k,\mu/k)$ converge to in distribution, when $k\to\infty$? $\endgroup$ – Did Nov 15 '16 at 7:48
  • $\begingroup$ @Did $S_k$ converges in distribution to Poisson($\mu$). I can see the idea of how this follows from $p_n \sim \mu/4^n$, but I have difficulty rigorously putting this down. Also, how can we get a.s. convergence from this? $\endgroup$ – takecare Nov 15 '16 at 7:58
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    $\begingroup$ For the convergence in distribution, use generating functions. For the almost sure convergence, note that $N_n$ increases to the number of points in $[0,1]$ (a fact which allows to reprove convergence in distribution, by the way). $\endgroup$ – Did Nov 15 '16 at 8:38

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