Maximal inequalities for additive processes (Q1930528)

A process \(X_{t}\), \(t\geq 0\), with independent increments, right-continuous with left-limits (rcll) paths, and values in \(\mathbb{R}^{d}\) is called additive if \(X_{t}\) is continuous in probability and \(X_{0}=0\). A nondecreasing and right-continuous function \(\phi (\cdot )\) with \(\phi (0)=0\), \(\phi (x)>0\), \(x>0\), is called moderate if there exist two constants \(\gamma \), \(\theta \in (0,\infty )\) such that \(\phi (x_{2})/\phi (x_{1})\leq \gamma (x_{2}/x_{1})^{\theta }\) whenever \(0<x_{1}<x_{2}<\infty \). As one of a number of results, the authors prove the following. Let \(X_{t}\) be an arbitrary additive process taking values in \(\mathbb{R}^{d}\), and let \(\phi (\cdot )\) be a moderate function. Consider \(X_{t}^{\ast }=\sup_{0\leq s\leq t}\left\| X_{s}\right\| \), where \(\left\| \cdot \right\| \) stands for Euclidean norm. A function \(a_{\phi }(\cdot )\) can be constructed in terms of the characteristics of \(X_{t}\) such that \(c_{1}\leq E\phi (X_{t}^{\ast })/Ea_{\phi }(t)\leq c_{2}\) holds uniformly for all \(t>0\), where \(0<c_{1}\leq c_{2}<\infty \) are two constants depending on \(\phi \) only.