Summary
Let {X t } be a one-dimensional Lévy process with local timeL(t, x) andL *(t)=sup{L(t, x): x ∈ ℝ}. Under an assumption which is more general than being a symmetric stable process with index α>1, we obtain a LIL forL*(t). Also with an additional condition of symmetry, a LIL for range is proved.
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Barlow, M.T.: Necessary and sufficient condition for the continuity of local times of Lévy processes. Ann. Probab.16, 1389–1427 (1988)
Barlow, M.T., Hawkes, J.: Application de l'entropie métrique a la continuité des temps locaux des processus de Lévy. C. R. Acad. Sci. Ser. I. Math.301, 237–239 (1985)
Borodin, A.N.: On the character of convergence to Brownian local time I. Probab. Theory Relat. Fields72, 231–250 (1986)
Bretagnolle, J.: Resultats de Kesten sur les processus a accroisements indépendants. In: Meyer, P.A. (ed.) Seminare de Probabilités V. (Lect. Notes Math., vol. 191, pp. 21–36) Berlin Heidelberg New York: Springer 1971
Getoor, R.K., Kesten, H.: Continuity of local times of Markov processes. Compos. Math.24, 277–303 (1972)
Griffin, P.S.: Probability estimates for the small deviations ofd-dimensional random walk. Ann. Probab.11, 939–952 (1983)
Griffin, P.S.: Laws of the iterated logarithm for symmetric stable processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.68, 271–285 (1985)
Jain, N.C., Pruitt, W.E.: Asymptotic behaivor of the local time of a recurrent random walk. Ann. Probab.12, 64–85 (1984)
Kesten, H.: An iterated logarithm law for local time. Duke Math. J.32, 447–456 (1965)
Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. Mem. Am. Math. Soc.93 (1969)
Pruitt, W.E.: General one-sided laws of the iterated logarithm. Ann. Probab.9, 1–48 (1981)
Ray, D.B.: Sojourn times of a diffusion process. Ill. J. Math.7, 615–630 (1963)
Wee, I.S.: Lower functions for processes with stationary independent increments. Probab. Theory Relat. Fields77, 551–566 (1988)
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This research is supported by a grant from Korea Science and Engineering Foundations
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Wee, IS. The law of the iterated logarithm for local time of a Lévy process. Probab. Th. Rel. Fields 93, 359–376 (1992). https://doi.org/10.1007/BF01193056
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DOI: https://doi.org/10.1007/BF01193056