发表机构
University of Bologna（博洛尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立随机加权均值的幂和收敛原理，结合相关定理证明了Breiman1965年猜想关于中心化可积标记的必要性方向，完善了该猜想。
AI 中文摘要
我们建立了随机加权均值的幂和收敛原理。设$P(t)=(P_j(t))$为随机有限支撑次概率权序列，与独立同分布的中心化可积标记无关。若期望总质量收敛，且某阶$r\in(0,1)$的期望幂和一致有界，则对固定标记律，$\sum_jP_j(t)X_j$收敛到非退化律将迫使$\mathbb{E}\sum_jP_j(t)^p$对某$p\in(1,2]$收敛到正极限。证明结合了特征函数余项Mellin变换的正态族紧性、频率$\pm u$处的反演，以及收敛横坐标处的Landau定理；一致阿贝尔估计处理端点$p=2$。对于由非负变量的泊松规模独立同分布样本归一化得到的权，拉普拉斯指数对数斜率小于1的缺口产生所需的阶小于1的幂和界。随后，泊松化比率陶伯型定理将未归一化变量的共同尾识别为正则变化，指数为$\beta\in[0,1)$。作为应用，这证明了Breiman1965年关于中心化可积标记的猜想中剩余的必要性方向，结合Breiman的充分性定理，该猜想得以完整证明。
英文摘要
We prove Breiman's conjecture under the first-moment assumption. Let $Y_1,Y_2,\ldots$ be iid nonnegative random variables with $\mathbb P\{Y_1>0\}>0$, 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 Poisson$\unicode{x2013}$Dirichlet 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 Fourier$\unicode{x2013}$Mellin identities extract a positive limiting expected power sum from one nondegenerate marked limit, without a moment of order greater than one. A ratio$\unicode{x2013}$Tauberian 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.
Comments65 pages, no figures. Retitled and reorganized, with expanded Breiman and subordinator results and transfer to all integrable marks and independently marked random measures. The $Λ$-coalescent results included in v2 are presented and further developed in a separate companion paper
