Variation inequalities for smartingales
Abstract
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.
Fields of Study & Subjects
Nonlinear Differential Equations Analysis
Stability and Controllability of Differential Equations
Advanced Mathematical Modeling in Engineering
Publication details
- Journal / source
- Arkiv för matematik
- Publisher
- International Press of Boston
- Volume
- 64
- Issue
- 1
- Pages
- 181–209
- Publication date
- 1 January 2026
- Language
- EN