Semiexponential Distributions and Related Large Deviation Principles for Trajectories of Random Walks

We obtain a rather simple characterization of semiexponential distributions. This allows us to relax substantially the conditions for the fulfillment of the moderately large deviation principle for the trajectories of random walks when Cramér’s condition does not hold. Besides, using the previous results, we establish the local large deviation principle (LDP) outside the zone of moderately large deviations for semiexponential random walks. The latter principle differs much from the LDP in the case that Cramér’s condition holds: The deviation function for it is concave but not convex, the deviation functional is finite only on jump trajectories, and so forth.