Maximal Inequalities for Additive Processes

Michael J. Klass () and Ming Yang ()
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Michael J. Klass: University of California, Berkeley
Ming Yang: Tulane University

Journal of Theoretical Probability, 2012, vol. 25, issue 4, 981-1012

Abstract: Abstract Let X t be an arbitrary additive process taking values in ℝ d . Consider $X_{t}^{*}=\sup_{0\le s\le t}\|X_{s}\|$ and a moderate function φ. We are able to construct a function a φ (t) from the characteristics of X t 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 T r =inf {t>0:‖X t ‖>r}, r>0. Similarly, we can define a function g φ (r) in terms of the characteristics of X t such that c 1 g φ (r)≤Eφ(T r )≤c 2 g φ (r) ∀r>0 for good constants c 1, c 2 depending only on φ.

Keywords: Additive processes; Lévy processes; Maximal inequalities; Stopping times; Random walks; Moderate functions; Sums of independent random variables; 60G40; 60E15; 60G51; 60G50 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10959-011-0357-4

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