= Power-law boundary accumulation of a Poisson point process
{title2=$n^{1/(\alpha-1)}(b-X_n)\to(c/(\alpha-1))^{1/(\alpha-1)}$}
If an <intensity function of a point process> has $\lambda(x)\sim c(b-x)^{-\alpha}$ at a finite endpoint, with $c>0$ and $\alpha>1$, the cumulative intensity satisfies $F(x)\sim c(b-x)^{1-\alpha}/(\alpha-1)$. The <Poisson mapping theorem> turns $F(X_n)$ into unit-rate arrivals. The <strong law for Poisson arrival times> gives $F(X_n)/n\to1$ <almost surely>, hence the displayed accumulation rate. Reversing the coordinate gives the same argument at a left endpoint. Different singularity exponents produce different powers of the index.
Back to article page
