AffiliationUniv Arizona, Dept Math
KeywordsLocal limit laws
Heavy-tailed random variables
Bivariate steepest descent
MetadataShow full item record
CitationErcolani, N.M., Jansen, S. & Ueltschi, D. J Theor Probab (2019) 32: 1. https://doi.org/10.1007/s10959-018-0832-2
Rights© Springer Science+Business Media, LLC, part of Springer Nature 2018
Collection InformationThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at firstname.lastname@example.org.
AbstractWe 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, Lindelof 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) and apply to logarithmic hazard functions cexp(-(logk)), >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.
Note12 month embargo; published online: 26 May 2018
VersionFinal accepted manuscript