Breiman's conjecture and normalized jumps of subordinators
Abstract
We prove Breiman's conjecture under the first-moment assumption. Let Y1,Y2,… be iid nonnegative random variables with ℙ{Y1>0}>0, normalized by their sum. If the resulting randomly weighted sum converges to a nondegenerate law for one fixed integrable, nonconstant mark distribution, then the tail of Y1 is regularly varying. More generally, any full-sequence limit for one such mark, including a constant limit, determines the asymptotic regime of the ranked weights: one big jump, a Poissonunicodex2013Dirichlet partition, or dust. It consequently determines the limit for every integrable mark, with convergence in the $1$-Wasserstein metric, and the limits of independently marked empirical measures. The inverse step is based on a countable power-sum theorem: signed Fourierunicodex2013Mellin identities extract a positive limiting expected power sum from one nondegenerate marked limit, without a moment of order greater than one. A ratiounicodex2013Tauberian argument then recovers the tail index. The same method classifies ratios formed from the marked jumps of a nonzero, unkilled, driftless subordinator at zero and at infinity, assuming infinite activity at zero. A nondegenerate limit is equivalent to regular variation of the Lévy tail with index in (−1,0]. A constant limit is equivalent to disappearance of the largest normalized jump, or, analytically, to slow variation of the integrated Lévy tail. The latter condition need not imply regular variation of the Lévy tail with index −1. A Cauchy-mark example shows that the conclusion can fail without integrability of the mark.
Versions
Related
- Subsequential Version of Breiman's Conjecture
- Noncommutative extensions of an ergodic theorem of Breiman
- Nonhomogeneous Subordinators and Lévy Copulas for Reliability Analysis of Time-Dependent Degradation Processes
- A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators
- Jumps and Iterated Jumps of Perfect Subtrees of Positive Trees
BibTeX
@misc{lenzi2026breimans,
title = {{Breiman's conjecture and normalized jumps of subordinators}},
author = {Jacopo Lenzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24616},
eprint = {2608.24616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24616},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Breiman's conjecture and normalized jumps of subordinators
AU - Jacopo Lenzi
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2608.24616
UR - https://arxiv.org/abs/2608.24616
AB - We prove Breiman's conjecture under the first-moment assumption. Let Y_1,Y_2,… be iid nonnegative random variables with ℙ{Y_1>0}>0, normalized by their sum. If the resulting randomly weighted sum converges to a nondegenerate law for one fixed integrable, nonconstant mark distribution, then the tail of Y_1 is regularly varying. More generally, any full-sequence limit for one such mark, including a constant limit, determines the asymptotic regime of the ranked weights: one big jump, a Poissonunicodex2013Dirichlet partition, or dust. It consequently determines the limit for every integrable mark, with convergence in the $1$-Wasserstein metric, and the limits of independently marked empirical measures. The inverse step is based on a countable power-sum theorem: signed Fourierunicodex2013Mellin identities extract a positive limiting expected power sum from one nondegenerate marked limit, without a moment of order greater than one. A ratiounicodex2013Tauberian argument then recovers the tail index. The same method classifies ratios formed from the marked jumps of a nonzero, unkilled, driftless subordinator at zero and at infinity, assuming infinite activity at zero. A nondegenerate limit is equivalent to regular variation of the Lévy tail with index in (−1,0]. A constant limit is equivalent to disappearance of the largest normalized jump, or, analytically, to slow variation of the integrated Lévy tail. The latter condition need not imply regular variation of the Lévy tail with index −1. A Cauchy-mark example shows that the conclusion can fail without integrability of the mark.
ER -