  Modulo the moment asymptotic expansion, the Cesaro and parametric behaviours
of distributions at infinity are equivalent. On the strength of this result, we
construct the asymptotic analysis for spectral densities, arising from elliptic
pseudodifferential operators. We show how Cesaro developments lead to efficient
calculations of the expansion coefficients of counting number functionals and
Green functions. The bosonic action functional proposed by Chamseddine and
Connes can more generally be validated as a Cesaro asymptotic development.
