Improper Poisson integral of the sinc function

A formal sum ${\sum }_{n}f(S_n)$ may be seen as the integral $\int fdN$ with respect to random point process $N(A)=\mathrm{\mid }\{n:S_n\in A\}\mathrm{\mid }$. We study its convergence beyond the well known context of Lebesgue integrable functions, admitting nonintegrable functions whose improper Riemann-Lebesgue integrals exist. We focus on the sinc function and some of its relations leaving the general case to conjectures.