Closure property of consistently varying random variables 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 a counting random variable independent of X. Consider randomly stopped sum S_η = ∑^η_{k = 1}X_k and random maximum S_(η) ? max?{S₀, …, S_η}. Assuming that each X_k belongs to the class of consistently varying distributions, on the basis of the well-known precise large deviation principles, we prove that the distributions of S_η and S_(η) belong to the same class under some mild conditions. Our approach is new and the obtained results are further studies of Kizinevi?, Sprindys, and ?iaulys (2016) and Andrulyt?, Manstavi?ius, and ?iaulys (2017).