Controlled Variable Tailed Independent Random Variables On The Heavy-tailed And Large Deviations

Since A.V.Nagaev and Heyde, many scholars deeply studied the large deviations for heavy-tailed random variables. These classical work mainly dealt with the claim size which are a sequence of independent and identically distributed random variables. Actually, the claim size usually are not identically distributed even not indpendent.In ref.[l], the authors studied the large deviations for partial and random sums of independent random variables with dominatedly varying tails, but the assumptions and the proof are not reasonable. In this paper, we further studied the large deviations for the partial and random sums especially the relevant centralizing sums, we have:{Xn, n > 1} are a sequence of independent non-negative random variables ,and {N(t), t > 0} is a process of non-negative integer-valued random variables , independent of {Xn ,n>1}. When the following conditions are satisfied:(A1) There exist a non-negative random variable X, with v = EX < and positive constant C1 such that for n large enough,(A2) There exist a non-negative random variable Y with its tail EY < and positive constant C2 such that for n large enough ,(A3)(A) We have :1. (1) for every fixed y > 0, (2) for every fixd 2.(1) for every fixed r>C2u, liminf inf (2) for every fixed r > C2u , 0 < m < 1limsup sup3.(1) for every fixed r>0, liminf inf (2) for every fixed r>0,0C2u, (2) for every fixed y >C2u,0