Maximal Inequalities for Additive Processes

Abstract

Let Xt be an arbitrary additive process taking values in ℝd. Consider \(X_{t}^{*}=\sup_{0\le s\le t}\|X_{s}\|\) and a moderate function φ. We are able to construct a function aφ(t) from the characteristics of Xt such that for all stopping times T, the ratio \(E\phi(X_{T}^{*})/Ea_{\phi}(T)\) is uniformly bounded away from 0 and ∞ by two constants depending on φ only. Let Tr=inf {t>0:‖Xt‖>r}, r>0. Similarly, we can define a function gφ(r) in terms of the characteristics of Xt such that c1gφ(r)≤Eφ(Tr)≤c2gφ(r) ∀r>0 for good constants c1, c2 depending only on φ.