Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

Let {i}i=1 be a sequence of independent identically distributed nonnegative random variables, Sn = ξ1 + ⋯ +ξn. Let Δ = (0, T] and x + Δ = (x, x + T]. We study the ratios of the probabilities P(Sn ε x + Δ)/P(ξ 1 ε x + Δ) for all n and x. The estimates uniform in x for these ratios are known for the so-called Δ-subexponential distributions. Here we improve these estimates for two subclasses of Δ-subexponential distributions; one of them is a generalization of the well-known class LC to the case of the interval (0, T] with an arbitrary T ≤ ∞. Also, a characterization of the class LC is given.