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Subsequential Version of Breiman's Conjecture

  • Volodymyr Pavlenkov
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Abstract

We study randomly weighted sums Tn=(∑i=1n XiYi)/(∑i=1n Yi), where (Xi) and (Yi) are independent sequences of i.i.d. random variables, with nonnegative weights Yi, 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 X1, by the failure of slow variation of the integrated tail, for weight Y1 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.

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Cite

BibTeXRIS
BibTeX
@misc{pavlenkov2026subsequential,
  title = {{Subsequential Version of Breiman's Conjecture}},
  author = {Volodymyr Pavlenkov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29551},
  eprint = {2610.29551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29551},
}
RIS
TY  - UNPB
TI  - Subsequential Version of Breiman's Conjecture
AU  - Volodymyr Pavlenkov
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29551
UR  - https://arxiv.org/abs/2610.29551
AB  - We study randomly weighted sums T_n=(∑_i=1^n X_iY_i)/(∑_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.
ER  -