In the paper's central model of independent, identically distributed observations, a full sequence of self-normalized ratios, averages divided by their own random totals, converges in distribution to a nondegenerate limit only when two conditions hold: the values being weighted, called marks, are nonconstant, and the denominator tail is regularly varying at infinity, meaning its large-value behavior follows a stable power-law pattern. The tail index is unique and lies from zero up to, but not including, one. The result links the visible limit to the hidden weighting mechanism.
The author presents this as the remaining necessity direction of Breiman's 1965 conjecture for centered, integrable marks. Combined with the sufficiency theorem, the paper says, the result completes that conjecture within the stated model. This is a result in theoretical probability, not an empirical study: the work describes random sequences, point processes and probability laws, but no participant sample.
The hidden condition in the weights
The proof reaches the iid conclusion through Poissonization, a ratio-Tauberian step and depoissonization. It also uses a strict logarithmic-slope gap and a uniform bound on fractional power sums of order below one, without imposing a moment assumption on the denominator variable. More broadly, the analysis uses Mellin transforms, Fourier inversion, normal-family compactness and Landau's theorem to control characteristic-function remainders.
The paper's broader power-sum principle starts with a uniform bound on the expected sum of fractional powers of the random weights, for an order strictly between zero and one. If the weighted mean has a nondegenerate limit, the expected power sum at some order above one and no greater than two must converge to a positive constant no greater than one. A power sum means adding the weights after raising each one to the chosen power. The result turns a distributional limit into a structural condition on the weights.
In the iid setting, that structural condition has a matching tail condition. A positive limiting expected power sum of order greater than one occurs exactly when the denominator tail is regularly varying at infinity, with a unique index from zero up to but not including one. The paper gives the limiting constant through a ratio of gamma functions involving the power order and the tail index. Tail behavior and weight concentration are therefore two descriptions of the same asymptotic regime in this model.
Two ways the limit can look
The limiting law is also specified. When the tail index is strictly between zero and one, it is a Poisson-Dirichlet weighted sum of independent and identically distributed marks, with the weights independent of those marks. At index zero, the limit is the mark itself. The two branches therefore describe either a random weighted combination or the one-big-jump endpoint described by the paper.
A parallel classification appears for ratios built from normalized Lévy jumps. Their nondegenerate endpoint branch requires a unique regular-variation index from zero up to but not including one for the Lévy-measure tail. The other, constant or dust, branch occurs when the largest jump becomes negligible relative to the truncated first-moment scale, and the marked ratio converges in probability to the mean mark. In that branch, the same condition is equivalent to the largest normalized weight tending to zero, every ranked weight tending to zero, expected power sums of orders above one vanishing, and slow variation of the integrated tail.
Beyond one ratio
The result extends from individual jump ratios to normalized random measures. For a nonconstant integrable mean functional, full-sequence endpoint convergence is equivalent to exactly one of two alternatives: a regularly varying Lévy tail or the maximal-jump condition behind the dust branch. The paper further states that both scalar alternatives lift to corresponding endpoint limits for the entire normalized random measure.
A separate theorem addresses fixed-time Lambda-coalescents. It says the probability law of one marked coalescent mean gives a topological embedding of the collision measure into real-valued probability laws. In practical terms, the collision measure is identified by the law of that one marked mean within this structured setting.
A mathematical result, not an empirical study
The conclusions are conditional on the assumptions built into the models, including independence and integrability, and the strongest classification statements concern full-sequence convergence. The paper does not report an empirical sample, so its findings are mathematical results rather than evidence from people or real-world interventions.
The supplied document is an arXiv version 2 preprint dated 30 August 2026. Its acknowledgments name personal encouragement and support from Marinella Mellini and Mauro Lenzi, but no research funder is named in the supplied text.
Paper data and sources
Original title: Power-sum convergence for randomly weighted means and Breiman's conjecture
Authors: Jacopo Lenzi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: 10.5281/zenodo.22083323
Original paper · Full text