On the Existence Conditions for Exact Large Deviation Principles

We consider large deviation principles (LDPs) for random walks and renewal compound processes. We expose and analyze the approaches to finding existence conditions for “exact” large deviation principles valid under minimal moment conditions. By exact large deviation principles we understand statements on the existence of exact limits for the asymptotics of the probabilities under study and the values of these limits rather than on the existence of upper and lower limits for these asymptotics, as for the ordinary large deviation principles. In addition to the known results, the new versions of conditions are found that ensure the existence of exact LDPs under the minimal Cramér’s moment condition. The article is based on the text of a talk prepared for a conference cancelled because of the pandemic.