Mathematicians proved the remaining necessity direction of Breiman 1965 conjecture by establishing a power sum convergence principle for randomly weighted means. Random weighted averages were not expected to force individual weight powers into strict positive limits to reach stable distributions. The proof shows that when random weights combine with centered values to form a non-degenerate law, the expected power sums of those weights must converge for an exponent between one and two.
When a sequence of random weights scales independent random variables, the total mass and lower power sums anchor the behavior of the system. The weights act like adjustable balance points on a scale where individual values cannot become too concentrated without upsetting the final distribution. The researchers combined normal family compactness for Mellin transforms with frequency inversion at positive and negative evaluation points. They applied Landau theorem at the abscissa of convergence while a uniform Abelian estimate resolved the upper boundary at exponent two.
The authors analyzed random finitely supported subprobability weight sequences paired with independent centered integrable random variables. For weights formed by normalizing a Poisson sample of non-negative variables, a slope gap below one in the Laplace exponent established a power sum bound under one. A Poisson ratio Tauberian theorem identified the shared tail of the unnormalized variables as regularly varying with an index between negative one and zero.
This convergence principle completes the full proof of Breiman 1965 conjecture when combined with the original sufficiency theorem. The result supplies the missing mathematical necessity requirement for centered integrable marks in weighted random sums.