 Symmetric inverse-power Poisson field on a line

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 $m\mathrm{sign}(x)/|x{|}^p$, 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.