  We derive a large deviation principle for families of random variables in the
basin of attraction of spectrally positive stable distributions by proving a
uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The
main advantage of this method is that it can be easily applied to cases that
are beyond the reach of the techniques currently used in the literature, such
as random walks with long-ranged memory kernels or randomly stopped sums where
the random time is not concentrated around its expectation. The method reveals
the role of the characteristic function when Cram\'er's condition is violated
and provides a unified approach within regular variation.
