Variation inequalities for smartingales

TL;DR

This paper extends a known inequality for martingales to smartingales, which are piecewise polynomial analogs, providing new bounds and estimates related to their asymptotic behavior.

Contribution

It introduces an extension of Makarov's inequality to smartingales on convex sets, including new estimates akin to the law of the iterated logarithm.

Findings

Extended inequality to smartingales on convex sets.

Established estimates similar to the law of the iterated logarithm.

Demonstrated the behavior of smartingales in the context of the inequality.

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 \[ \liminf_{n\to\infty} \frac{M_n}{\sum_{k=1}^n |\Delta M_k|} > 0 \] 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.