Upper-bound estimates for weighted sums satisfying Cramer’s condition

Abstract

Let S = ω1S1 + ω2S2 + ⋯ + ωNSN. Here Sj is the sum of identically distributed random variables and ωj > 0 denotes weight. We consider the case, when Sj is the sum of independent random variables satisfying Cramer’s condition. Upper-bounds for the accuracy of compound Poisson first and second order approximations in uniformmetric are established.