Subsequential Version of Breiman's Conjecture

We study randomly weighted sums $$T_n=\frac{\sum _{i=1}^n{X}_i{Y}_i}{\sum _{i=1}^n{Y}_i},$$ where $(X_i)$ and $(Y_i)$ are independent sequences of i.i.d. random variables, with nonnegative weights $Y_i$, motivated by Breiman’s conjecture. We introduce the class $ISSV$ of $I$-subsequentially slowly varying tails and establish a subsequential analogue of Darling’s maximum–sum theorem. This provides a subsequential generalization of Breiman’s slowly varying theorem and, as a corollary, resolves the slowly varying case of his conjecture. We also establish a subsequential analogue of Breiman’s nonzero-index theorem, characterizing the existence of nondegenerate subsequential limits, distinct in distribution from $X_1,$ by the failure of slow variation of the integrated tail, for weight $Y_1$ with tail distributions outside the class $ISSV$. Finally, we formulate the subsequential separation principle, prove it under a finite-second-moment assumption, and show that its general validity would imply the nonzero-index case of Breiman’s conjecture.