Asymptotics For Sums Of Random Variables With Local Subexponential

In this thesis, we study the tail behaviour of random sum Z=(?) Xi,where X1, X2,…is a sequence of independent identically distributed random variables with common distribution F and v is an integer valued random variable.In this paper, we obtain the asymptotic behaviour of the random sum Z, our result is similar to that of Kalma(1972) and Asmussen(2003), but the conditions on v is weakened. In addition, we also generalize the results of Daley,Omey,Vesilo(2007) to a more widely distribution class.In chapter one, some important heavy-tailed distribution classes are reviewed and main results in the thesis are presented. We obtain local asymptoties for the sums of random variables under a weaker condition Ev<∞than that of Asmussen(2003) E(1+ε)"< 00. In addition, we generate the results of Daley,Omey and Vesilo(2007) from the conditions that X1 belongs to S and D∩C to that X1 belongs to S(γ) and D∩(?)(γ). The proofs of our results are postponed to chapters two and three.