I am struggling to solve the following problem as part of a bigger project that I am working on.

Let $\mathcal{S} \subset \mathbb{R}^2$ be a square of length $\sqrt{n}$ centered at the origin, $f:\mathcal{S} \times \mathcal{S} \rightarrow \mathbb{R}$ given by $f(x,y) = e^{-|x-y|}|x-y|^2*\Big( \frac{1}{\max(\min(|x|,|y|),1)} \Big)^2$.

I want to show that $\frac{1}{(\log n)^2} \int_{\mathcal{S}x \mathcal{S}} f(x,y) dx dy$ converges to $0$ as $n$ goes to infinity. Notice that here $x,y \in \mathcal{S}$, so they are basically 2-d vectors.

Using Campbell-Hardy theorem, this is equivalent to showing that the expected value of the sum of $f(x,y)$ converges to $0$, where $x,y$ run over all the points in a Poisson Point Process of rate $1$ over this square. Using Python, I proved empirically that the claim above is true. Now, I am trying to find a rigorous math proof.

I tried breaking the integral in easier to compute integrals depending on the positions of $x,y$, but all of my approaches were futile so far.

  • $\begingroup$ Are you sure $f$ does not depend on $n$? As it stands, the integral over that square is nondecreasing with $n$. $\endgroup$ – punctured dusk Mar 14 at 20:25
  • $\begingroup$ I forgot a term for the limit part. Thanks for bringing this up, please check the updated version. $\endgroup$ – user548645 Mar 14 at 20:27

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