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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.13227 |
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Table of Contents:
- A result by N.G. Makarov [Algebra i Analiz, 1989] states that for martingales $(M_n)$ on the torus we have the strict inequality \[ \liminf_{n\to\infty} \frac{M_n}{\sum_{k=1}^n |ΔM_k|} > 0 \] on a set of Hausdorff dimension one, denoting by $ΔM_n$ the martingale differences $ ΔM_n = M_n - M_{n-1} $. We discuss an extension of this inequality to so-called smartingales on convex, compact subsets of $\mathbb R^d$, which are piecewise polynomial (or spline) versions of martingales. As a tool we need and prove an estimate for smartingales in the spirit of the law of the iterated logarithm.