Almost Supermartingale Extensions of Olivier's Theorem

Patrick L. Combettes; Javier I. Madariaga

Almost Supermartingale Extensions of Olivier's Theorem

Probability 2026-07-02 v1 Optimization and Control

Authors: Patrick L. Combettes , Javier I. Madariaga

Abstract

Olivier's 1827 theorem provides a rate of convergence to zero of the general term of a decreasing summable sequence of positive reals. We derive stochastic extensions of this result in the context of almost supermartingales. The results are applied to the analysis of stochastic iterative processes.

Keywords

Olivier's theorem almost supermartingale stochastic extension stochastic iterative process convergence rate

Cite

@article{arxiv.2607.02489,
  title  = {Almost Supermartingale Extensions of Olivier's Theorem},
  author = {Patrick L. Combettes and Javier I. Madariaga},
  journal= {arXiv preprint arXiv:2607.02489},
  year   = {2026}
}