Closure property of random sum and its maximum of random variables from class D based on precise large deviation principles

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.