Probability inequalities for sums of independent unbounded random variables

The tail probability inequalities for the sum of independent unbounded rand om variables on a probability space (Omega, T, P) were studied and a new me thod was proposed to treat the sum of independent unbounded random variable s by truncating the original probability space (Omega, T, P). The probabili ty exponential inequalities for sums of the results, some independent unbou nded random variables were given. As applications of interesting examples w ere 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 prop erties of the sums of independent unbounded random variables.