  A power counting rule is provided that allows us to obtain upper bounds for
the absolute values of Feynman parametric integrands. The rule reflects both
the ultraviolet and infrared behavior taking into account that the external
momenta are on the mass shell. It gives us the ability to rigorously prove the
absolute convergence of the corresponding integrals. The consideration is
limited to the case of the quantum electrodynamics Feynman graphs contributing
to the lepton magnetic moments and not containing either lepton loops or
ultraviolet divergent subgraphs. However, a rigorous mathematical proof is
given for all Feynman graphs satisfying these restrictions. The power counting
rule is formulated in terms of Hepp's sectors, ultraviolet degrees of
divergence and so-called I-closures. The obtained upper bound can not be
substantially improved: the illustrative example is provided. The paper
provides the first mathematically rigorous treatment of the ultraviolet
behavior together with the on-shell infrared behavior with some kind of
generality. Practical applications of this rule are explained.
