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, S _n = ?₁ + ? +?_n. Let ? = (0, T] and x + ? = (x, x + T]. We study the ratios of the probabilities P(S _n e x + ?)/P(? ₁ e x + ?) for all n and x. The estimates uniform in x for these ratios are known for the so-called ?-subexponential distributions. Here we improve these estimates for two subclasses of ?-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 8. Also, a characterization of the class LC is given. © 2006 Springer Science+Business Media, Inc.