Abstarct: Abstract In this thesis, the necessary and sufficient conditions for complete convergence and Kolmogorov's law for sums A weight of negative dependent random variables -expanded ⅿ is prepared. Some applications of the main results have also been suggested. including strong and weak compatibility estimates Least Squares in Multiple Linear Regression Model and Robust Adaptation (Strong Convergence) Value Estimation Conditional risk (VaR) (at the end some numerical simulations to confirm the results A comment is made.