Summary
A necessary and sufficient Fourier analytic condition is given for a real-valued Lévy process X to be such that all perturbations, X+f, of X by measurable functions f have range with positive Lebesgue measure. The condition is compared to one due to Kesten for the range of X alone to have positive Lebesgue measure, and in some cases it is found to coincide, whereas in others it is stronger.
References
Bretagnolle, J.: Résultats de Kesten sur les processus à accroissements indépendants. In: Séminaire de probabilités V, Strasbourg 1971. (Lect. Notes Math., vol. 191) Berlin Heidelberg New York: Springer 1971
Kahane, J-P.: Some random series of functions. Lexington: Heath 1968
Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. Mem. Am. Math. Soc. vol.93. Providence: AMS 1969
Mountford, T.: Time inhomogeneous Markov processes and the polarity of single points. Ann. Probab., in press
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Partially supported by NSF grant DMS 8701212
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Evans, S.N. The range of a perturbed Lévy process. Probab. Th. Rel. Fields 81, 555–557 (1989). https://doi.org/10.1007/BF00367302
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DOI: https://doi.org/10.1007/BF00367302
Keywords
- Fourier
- Stochastic Process
- Probability Theory
- Measurable Function
- Statistical Theory