Solution

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 28 / 5 / c / Solution

{title2=$\phi(t)=\exp\{-\rho\pi m^\alpha|t|^\alpha/[\Gamma(\alpha)\sin(\pi\alpha/2)]\}$}

Place equal masses at points of a homogeneous <Poisson point process> of intensity $\rho$ on the line, and truncate their signed inverse-power forces symmetrically at radius $R$. For force $m\operatorname{sign}(x)/|x|^p$, with $\alpha=1/p\in(0,2)$, the characteristic exponent tends to $-2\rho\int_0^\infty(1-\cos(mt/x^p))\,dx$. The <cosine integral for a symmetric stable exponent> evaluates it as displayed. Thus the weak limit is a <symmetric stable distribution>. Symmetric truncation is important when the far-field force series is not absolutely summable.

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 Incoming links: Symmetric inverse-power Poisson field on a line