Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Abstract

Let {Xn; n ≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E(X1) and let {N(t); t ≥ 0} be a random process taking non-negative integer values with finite mean λ(t) = E(N(t)) and independent of {Xn; n ≥ 1}. In this paper, asymptotic expressions of P((X1+⋯+XN(t))−λ(t)μ > x) uniformly for x ∈ [γb(t),∞) are obtained, where γ > 0 and b(t) can be taken to be a positive function with limt→∞ b(t)/λ(t) = 0.