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 $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{M_n}{\sum _{k=1}^n\mathrm{\mid }\mathrm{\Delta }{M}_k\mathrm{\mid }}>0$$ on a set of Hausdorff dimension one, denoting by $\mathrm{\Delta }{M}_n$ the martingale differences $\mathrm{\Delta }{M}_n=M_n-M_{n-1}$. 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.