Breiman’s 1965 conjecture

Let $(X_i)_{i\ge1}$ be iid centered integrable random variables, let $(Y_i)_{i\ge1}$ be iid nonnegative random variables independent of $(X_i)$ , and set $S_n=\sum_{i=1}^nY_i$ . If, for some nondegenerate random variable $Z$ , $\sum_{i=1}^n\frac{Y_i}{S_n}X_i\Rightarrow Z$ , then the conjecture asserts that there exists $\alpha\in[0,1)$ such that $\mathbb{P}(Y_1>x)$ is regularly varying at infinity with index $-\alpha$ ; equivalently, the law of $Y_1$ belongs to the domain of attraction $D(\alpha)$ of a positive stable law of index $\alpha$ .

References

Primary source

arXiv

Additional references

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed preprint claims to settle Breiman’s conjecture for centered integrable variables, but the result has not been independently verified.

Breiman’s 1965 conjecture says that convergence of one normalized randomly weighted sum to a nondegenerate limit should force the weights to have a specific heavy-tail domain-of-attraction property. Breiman proved the converse under the relevant hypotheses.

Known results

Breiman, 1965: established the sufficiency direction for weights in the domain of attraction $D(\alpha)$ , with $0\le\alpha<1$ .
Mason and Zinn, 2005: proved necessity when the centered integrable mark satisfies $\mathbb{E}|X|^p<\infty$ for some $p>2$ .
A 2015 preprint extended necessity to a specified class including finite-variance marks and some stable-domain cases, not all integrable marks.

August 2026 preprint

On August 25, 2026, Power-sum convergence for randomly weighted means and Breiman's conjecture claimed the missing necessity condition in the centered-integrable setting and combined it with Breiman’s theorem to claim completion. The claim is based on an unrefereed preprint and is therefore unverified.

Current status (as of August 2026): Breiman’s conjecture is claimed solved for the centered-integrable setting, while independent verification and the precise scope of the preprint’s hypotheses remain open.

Sources

No solutions have been posted yet.