Probability inequalities for sums of independent unbounded random variables

The tail probability inequalities for the sum of independent unbounded random variables on a probability space (Omega, T, P) were studied and a new method was proposed to treat the sum of independent unbounded random variables by truncating the original probability space (Omega, T, P). The probability exponential inequalities for sums of the results, some independent unbounded random variables were given. As applications of interesting examples were given. The examples show that the method proposed in the paper and the results of the paper are quite useful in the study of the large sample properties of the sums of independent unbounded random variables.