On probabilities of large deviations for random walks. II. Regular exponentially decaying distributions

We establish exact asymptotic behavior for the probabilities of crossing arbitrary curvilinear boundaries in the large deviations range by random walks, whose jump distribution tails differ from an exponential function by an integrable regularly varying factor. In this interesting transient case, there exists a “lower subzone" of the zone of large deviations, where the classical exact asymptotic results hold true, and an "upper subzone,” where only results on the crude logarithmic asymptotics were available. Now we derive exact asymptotic behavior for the latter subzone and show that it is, in a sense, close to that described in the first part of the paper [Theory Probab. Appl., 46 (2001), pp. 193–213], where we dealt with regularly varying distribution tails. Moreover, under an additional "asymptotic smoothness" condition on the jumps distribution, we establish an asymptotic expansion for the tails of the distributions of the sums of the jumps in the large deviations range.