Abstarct: Three important type of statistical distributions family, family heavy-tailed distributions, long-tailed and are subexponential. The largest of these families, the family of heavy-tailed distributions. In this thesis, at first express distribution arrangements and the definitions of these three families and bring proven results. In addition, in the second chapter the definition of family to give the dominatedly varying-tailed distributions and show the family class is closure under the action of convolutions long-tailed and dominatedly varying-tailed distributions. Finally, the last chapter of this thesis by considering the a weighted random variables X_1 and X_2 and create a certain dependence structure between these two variables‏, is proof Close family ‎long-tailed ‎‎and dominatedly varying-tailed distributions under convolutions and expressed planned equivalence results.‎‎