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Some independence results related to the arc-sine law. (English) Zbl 0876.60060
This paper is devoted to the proof of recent results of Getoor and Sharpe on the distribution of local times on rays for planar Lévy processes $(X,Y)$. The main interest is on the local time $l$ of $Y$ and the process $L_t=\int^t_0 1_{\{X_s>0\}}dl_s$. It is proved that for every time $t>0$, $l_t$ and $L_t/l_t$ are independent and the latter is distributed according to a generalized arc-sine law. An asymptotic result concerning the law of $L_t$ is proved as well.

60J55Local time, additive functionals
Full Text: DOI
[1] Bertoin, J., and Doney, R. A. (1995). Spitzer’s condition for random walks and Lévy processes. Preprint. · Zbl 0880.60078
[2] Darling, D. A., and Kac, M. (1957). On occupation-times for Markov processes.Trans. Amer. Math. Soc. 84, 444--458. · Zbl 0078.32005 · doi:10.1090/S0002-9947-1957-0084222-7
[3] Doney, R. A. (1993). A path decomposition for Lévy processes.Stoch. Proc. Appl. 47, 167--181. · Zbl 0779.60046 · doi:10.1016/0304-4149(93)90012-S
[4] Doney, R. A. (1995). Spitzer’s condition and ladder variables in random walks.Prob. Th. Rel. Fields 101, 577--580. · Zbl 0818.60060 · doi:10.1007/BF01202785
[5] Fristedt, B. E. (1974). Sample functions of stochastic processes with stationary independent increments. In Ney, P., and Port, S. (eds.),Advances in Applied Probability III, Dekker, pp. 241--396. · Zbl 0309.60047
[6] Getoor, R. K., and Sharpe, M. J. (1994). On the Arc-sine laws for Lévy processes.J. Appl. Prob. 31, 76--89. · Zbl 0802.60070 · doi:10.2307/3215236
[7] Getoor, R. K., and Sharpe, M. J. (1994). Local times on rays for a class of planar Lévy processes.J. Th. Prob. 7, 799--811. · Zbl 0807.60071 · doi:10.1007/BF02214373
[8] Greenwood, P., and Pitman, J. (1980). Fluctuation identities for Lévy processes and splitting at the maximum.Adv. Appl. Prob. 12, 893--902. · Zbl 0443.60037 · doi:10.2307/1426747
[9] Kesten, H. (1969). Hitting probabilities of single points for processes with stationary independent increments.Mem. Amer. Math. Soc. Vol. 93. · Zbl 0201.19002
[10] Lebedev, N. (1972).Special Functions and their Applications. Dover. · Zbl 0271.33001
[11] Molchanov, S. A., and Ostrovskii, E. (1969). Symmetric stable processes as traces of degenerate diffusion processes.Th. Prob. Appl. XIV-1, 128--131. · Zbl 0281.60091 · doi:10.1137/1114012
[12] Pecherskii, E. A., and Rogozin, B. A. (1969). On joint distributions of random variables associated with fluctuations of a process with independent increments.Th. Prob. Appl. 14, 410--423. · doi:10.1137/1114054