A Local Limit Theorem for Large Deviations

A local limit theorem for large deviations for the i.i.d. random variables was given by Richter (1957), who used the saddle point method of complex variables to prove it. In this paper we give an alternative form of local limit theorem for large deviations for the i.i.d. random variables which is essentially equivalent to that of Richter. We prove the theorem by more direct and heuristic method under a rather simple condition on the moment generating function (m.g.f.). The theorem is proved without assuming that $E(X_i)=0$.