Integro-local and integral theorems for sums of random variables with semiexponential distributions

We obtain some integro-local and integral limit theorems for the sums S(n) = xi + center dot center dot center dot + xi (n) of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form P(xi >= t) = e(-t beta L(t)), where beta is an element of (0, 1) and L(t) is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as x -> infinity of the probabilities P (S(n) is an element of [x, x + Delta)) and P (S(n) >= x) in the zone of normal deviations and all zones of large deviations of x: in the Cramer and intermediate zones, and also in the "extreme" zone where the distribution of S(n) is approximated by that of the maximal summand.