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

Let {xi(i)}(i=1) be a sequence of independent identically distributed nonnegative random variables, S-n=xi(1)+...+epsilon(n). Let Delta=(0, T] and x+Delta=(x, x+T]. We study the ratios of the probabilities P(S-n is an element of x+Delta)/P(xi(1) is an element of x+Delta) for all n and x. The estimates uniform in x for these ratios are known for the so-called Delta-subexponential distributions. Here we improve these estimates for two subclasses of Delta-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 <=infinity. Also, a characterization of the class LC is given.