Scale invariance of uniform distribution over $\mathbb R^2$?

Question

If we make a uniform distribution of points over $\mathbb R^2$ with 1 point on average per unit square. And we zoom far out and make a density plot (give a color to each cell according to how many Points it contains, normalized to average) and we zoom even farther out and do the same I expect it to have the same distribution. Is this true? What distribution is it? Is there a way to make precise and prove this scale inariance?

Answer 1

I'm guessing that what you have in mind is a Poisson process. Two essential facts about the process described in your question are the following.

• Let $X$ be the number of sites falling within a particular region whose area is $a$. Then $\operatorname{E}(X) = a.$
• If two regions do not intersect each other, then the numbers of sites falling within them are independent random variables.

From those facts it is possible to deduce that $\Pr(X = n) = \dfrac{a^n e^{-a}}{n!}$.

This discrete probability distribution is a Poisson distribution.