Singularity Analysis for Heavy-Tailed Random Variables

We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindelöf integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by Nagaev (Transactions of the sixth Prague conference on information theory, statistical decision functions, random processes, Academia, Prague, 1973). The theorems generalize five theorems by Nagaev (Litov Mat Sb 8:553–579, 1968) on stretched exponential laws p(k) = cexp (- k^α) and apply to logarithmic hazard functions cexp (- (log k) ^β) , β> 2 ; they cover the big-jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem.