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Maximal Inequalities for Additive Processes

Author

Listed:
  • Michael J. Klass

    (University of California, Berkeley)

  • Ming Yang

    (Tulane University)

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.

Suggested Citation

  • Michael J. Klass & Ming Yang, 2012. "Maximal Inequalities for Additive Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 981-1012, December.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:4:d:10.1007_s10959-011-0357-4
    DOI: 10.1007/s10959-011-0357-4
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    References listed on IDEAS

    as
    1. Novikov, Alexander & Valkeila, Esko, 1999. "On some maximal inequalities for fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 47-54, August.
    2. De La Peña, Victor H. & Yang, Ming, 2004. "Bounding the first passage time on an average," Statistics & Probability Letters, Elsevier, vol. 67(1), pages 1-7, March.
    3. Yang, Ming, 2002. "Occupation times and beyond," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 77-93, January.
    4. Goran Peskir, 2001. "Bounding the Maximal Height of a Diffusion by the Time Elapsed," Journal of Theoretical Probability, Springer, vol. 14(3), pages 845-855, July.
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