Abstract: Let X={X1,X2,…} be a sequence of independent but not necessarily identically distributed random variables,and let η be an integer-valued counting random variable independent of X .Random sum Sη=∑ηk=1 Xk and its maximum S(η)=max{S0,…,Sη} were studied. Assuming that each Xk belongs to the class of D , by using the result of the precise large deviation principles on class D , it was proven that the distributions of Sη and S(η) belong to the same class under some conditions.The obtained results expand the related ones of previous studies.

Key words: random sum, random maximum, precise large deviation, closure property