  The constraints imposed by asymptotic freedom and analyticity on the
large-order behavior of perturbation theory for the electromagnetic
current-current correlation function are examined. By suitably applying the
renormalization group, the coefficients of the asymptotic expansion in the deep
Euclidean region may be expressed explicitly in terms of the perturbative
coefficients of the Minkowski space discontinuity (the $R$-ratio in $e^+ e^-$
scattering). This relation yields a ``generic'' prediction for the large-order
behavior of the Euclidean perturbation series and suggests the presence of
non-perturbative $1/q^2$ correction in the Euclidian correlation function. No
such ``generic'' prediction can be made for the physically measurable
$R$-ratio. A novel functional method is developed to obtain these results.
