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

Non-Improved Uniform Tail Estimates for Normed Sums of Independent Random Variables with Heavy Tails, with applications