  We obtain an uniform tail estimates for natural normed sums of independent
random variables (r.v.) with regular varying tails of distributions.
  We give also many examples on order to show the exactness of offered
estimates and discuss some applications in the method Monte-Carlo and
statistics, and obtain the sufficient conditions for Central and stable limit
theorem in the Banach space of continuous function.
  There are considered a slight generalization on a random variables with
super-heavy tails and martingale difference scheme.
