Power-sum convergence for randomly weighted means and Breiman's conjecture

Abstract

We establish a power-sum convergence principle for randomly weighted means. Let $P(t)=(P_j(t))$ be random finitely supported subprobability weight sequences, independent of iid centered integrable marks. If the expected total mass converges and the expected power sum of some order $r\in(0,1)$ is uniformly bounded, then convergence, for one fixed mark law, of $\sum_jP_j(t)X_j$ to a nondegenerate law forces $\mathbb{E}\sum_jP_j(t)^p$ to converge to a positive limit for some $p\in(1,2]$. The proof combines normal-family compactness for Mellin transforms of characteristic-function remainders, inversion at frequencies $\pm u$, and Landau's theorem at the abscissa of convergence; a uniform Abelian estimate handles the endpoint $p=2$. For weights obtained by normalizing a Poisson-sized iid sample of nonnegative variables, a gap below one for the logarithmic slope of the Laplace exponent yields the required power-sum bound of order below one. A Poissonized ratio Tauberian theorem then identifies the common tail of the unnormalized variables as regularly varying, with index $-\beta$ for a unique $\beta\in[0,1)$. As an application, this proves the remaining necessity direction in Breiman's 1965 conjecture for centered integrable marks. Combined with Breiman's sufficiency theorem, it completes the conjecture.