Variation inequalities for smartingales

A result by N.G. Makarov [Algebra i Analiz, 1989] states that for martingales \((M_n)\) on the torus we have the strict inequality \[ \liminf_{n\to\infty} \frac{M_n}{\sum_{k=1}^n |\Delta M_k|} > 0 \] on a set of Hausdorff dimension one, denoting by \(\Delta M_n\) the martingale differences \( \De...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Author: Passenbrunner, Markus
Format: Article
Language:English
Subjects:
Martingales
Polynomials
Set theory
Toruses
Online Access:Get full text
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Summary:A result by N.G. Makarov [Algebra i Analiz, 1989] states that for martingales \((M_n)\) on the torus we have the strict inequality \[ \liminf_{n\to\infty} \frac{M_n}{\sum_{k=1}^n |\Delta M_k|} > 0 \] on a set of Hausdorff dimension one, denoting by \(\Delta M_n\) the martingale differences \( \Delta M_n = M_n - M_{n-1} \). 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.
ISSN:2331-8422