A sharp inequality for the tail probabilities of sums of i.i.d. r.v.’s with dominatedly varying tails

Abstract

LetF be a distribution function supported on (-∞,∞) with a finite mean μ. In this note we show that if its tail$\overline F = 1 - F$ is dominatedly varying, then for any γ > max{μ, 0}, there exist C(γ) > 0 and D(γ) > 0 such that

$$C(\gamma )n\overline F (x) \leqslant \overline {F^{n * } } (x) \leqslant D(\gamma )\overline F (x),$$

for all n ≥ 1 and all x ≥ γn. This inequality sharpens a classical inequality for the subexponential distribution case