Abstract: We have discovered the relationship between classical heavy-tailed class D of random variables and heavy-tailed class ERV which can be more easily dealt with，and we have obtained the accurate large deviation conclusion about random sums of heavy-tailed random variables that belongs to the class D. As to the random sums when summands’distribution functions are heavier than λ·e^-c , we have got similar conclusions.

Key words: heavy-tailed random variable, accurate large deviation, moment generating function, subexponential distribution