Thetailprobabilityinequalitiesforthesumofindependentunboundedrandomvariablesonaprobabilityspace(Ω,T,P)werestudiedandanewmethodwasproposedtotreatthesumofindependentunboundedrandomvariablesbytruncatingtheoriginalprobabilityspace(Ω,T,P).Theprobabilityexponentialinequalitiesforsumsofindependentunboundedrandomvariablesweregiven.Asapplicationsoftheresults,someinterestingexamplesweregiven.Theexamplesshowthatthemethodproposedinthepaperandtheresultsofthepaperarequiteusefulinthestudyofthelargesamplepropertiesofthesumsofindependentunboundedrandomvariables.
