Abstract
Let X t 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 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.
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References
De la Peña, V.H.: From boundary crossing of nonrandom functions to first passage times of processes with independent increments. Unpublished manuscript (1997)
De la Peña, V.H., Eisenbaum, N.: Exponential Burkholder Davis Gundy inequalities. Bull. Lond. Math. Soc. 29, 239–242 (1997)
De la Peña, V.H., Giné, E.: Decoupling: From Dependence to Independence. Springer, New York (1999)
De la Peña, V.H., Yang, M.: Bounding the first passage time on an average. Stat. Probab. Lett. 67, 1–7 (2004)
Graverson, S.E., Peskir, G.: Maximal inequalities for the Ornstein–Uhlenback process. Research Reports No. 393, Dept. Theoret. Statist. Aarhus (1998)
Hahn, M.G., Klass, M.J.: Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal. Ann. Probab. 8, 262–280 (1980)
He, S.W., Wang, J.G., Yan, J.A.: Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe (Science Press), Beijing (1992)
Klass, M.J.: Toward a universal law of the iterated logarithm, Part I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 36, 165–178 (1976)
Klass, M.J.: A method of approximating expectations of functions of sums of independent random variables. Ann. Probab. 9, 413–428 (1981)
Klass, M.J.: Uniform lower bounds for randomly stopped Banach space valued random sums. Ann. Probab. 18, 780–809 (1990)
Klass, M.J., Nowicki, K.: An optimal bound on the tail distribution of the number of recurrences of an event in product spaces. Probab. Theory Relat. Fields 126, 51–60 (2003)
Novikov, A., Valkeila, E.: On some maximal inequalities for fractional Brownian motions. Stat. Probab. Lett. 44, 47–54 (1999)
Peskir, G.: Bounding the maximal height of a diffusion by the time elapsed. J. Theor. Probab. 14, 845–855 (2001)
Pruitt, W.E.: The growth of random walks and Lévy processes. Ann. Probab. 9, 948–956 (1981)
Yang, M.: Occupation times and beyond. Stoch. Process. Appl. 97, 77–93 (2002)
Yang, M.: The growth of additive processes. Ann. Probab. 35, 773–805 (2007)
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Klass, M.J., Yang, M. Maximal Inequalities for Additive Processes. J Theor Probab 25, 981–1012 (2012). https://doi.org/10.1007/s10959-011-0357-4
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DOI: https://doi.org/10.1007/s10959-011-0357-4
Keywords
- Additive processes
- Lévy processes
- Maximal inequalities
- Stopping times
- Random walks
- Moderate functions
- Sums of independent random variables