Breiman's conjecture and normalized jumps of subordinators

Abstract

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 L\'evy 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 L\'evy tail. The latter condition need not imply regular variation of the L\'evy tail with index $-1$. A Cauchy-mark example shows that the conclusion can fail without integrability of the mark.