Logarithmic Large Deviations for Heavy-Tailed Randomly Stopped Sums
Abstract
We derive logarithmic large-deviation asymptotics for randomly stopped sums with nonnegative summands satisfy a weak form of regular variation. Our approach is based on a count-scale condition that controls the logarithmic order of the random number of summands and does not require relative concentration around its expectation. A main advantage of the framework is that it applies to every renewal count, without moment, regular-variation, or domain-of-attraction assumptions on the interarrival distribution. In particular, it covers infinite-mean interarrival regimes in which relative concentration may fail. It is proven that the mean renewal count is an admissible count scale even when relative concentration fails. We further apply the results to finite heterogeneous insurance portfolios, obtaining dependence-robust asymptotics for unions and weighted aggregate losses, as well as explicit joint shortfall costs under block independence.
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BibTeX
@misc{zapata2026logarithmic,
title = {{Logarithmic Large Deviations for Heavy-Tailed Randomly Stopped Sums}},
author = {José M. Zapata},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.16487},
eprint = {2606.16487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16487},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithmic Large Deviations for Heavy-Tailed Randomly Stopped Sums
AU - José M. Zapata
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.16487
UR - https://arxiv.org/abs/2606.16487
AB - We derive logarithmic large-deviation asymptotics for randomly stopped sums with nonnegative summands satisfy a weak form of regular variation. Our approach is based on a count-scale condition that controls the logarithmic order of the random number of summands and does not require relative concentration around its expectation. A main advantage of the framework is that it applies to every renewal count, without moment, regular-variation, or domain-of-attraction assumptions on the interarrival distribution. In particular, it covers infinite-mean interarrival regimes in which relative concentration may fail. It is proven that the mean renewal count is an admissible count scale even when relative concentration fails. We further apply the results to finite heterogeneous insurance portfolios, obtaining dependence-robust asymptotics for unions and weighted aggregate losses, as well as explicit joint shortfall costs under block independence.
ER -