The Annals of Probability

Stability of the Overshoot for Lévy Processes

R.A. Doney and R.A. Maller

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We give equivalences for conditions like $X(T(r))/r\rightarrow 1$ and $X(T^{*}(r))/\allowbreak r\rightarrow 1$, where the convergence is in probability or almost sure, both as $r\rightarrow 0$ and $r\rightarrow \infty$, where $X$ is a L\'{e}vy process and $T(r)$ and $T^{*}(r)$ are the first exit times of $X$ out of the strip $\{(t,y):t> 0,|y|\leq r\}$ and half-plane $\{(t,y):t> 0$, $y\leq r\}$, respectively. We also show, using a result of Kesten, that $X(T^{*}(r))/r\rightarrow 1$ a.s.\ as $r\to 0$ is equivalent to $X$ ``creeping'' across a level.

Article information

Ann. Probab., Volume 30, Number 1 (2002), 188-212.

First available in Project Euclid: 29 April 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G17: Sample path properties

Processes with independent increments exit times first passage times local behavior


Doney, R.A.; Maller, R.A. Stability of the Overshoot for Lévy Processes. Ann. Probab. 30 (2002), no. 1, 188--212. doi:10.1214/aop/1020107765.

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  • BERTOIN, J. (1996). An Introduction to Lévy Processes. Cambridge Univ. Press.
  • BERTOIN, J. (1997). Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121 345-354.
  • BERTOIN, J. and DONEY, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Probab. Lett. 21 363-365.
  • CHOW, Y. S. (1986). On moments of ladder height variables. Adv. Appl. Math. 7 46-54.
  • DONEY, R. A. and MALLER, R. A. (2002). Stability and attraction to Normality for Lévy processes at zero and infinity. J. Theoret. Probab. To appear.
  • FELLER, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • GRIFFIN, P. S. and MALLER, R. A. (1998). On the rate of growth of the overshoot and the maximal partial sum. Adv. Appl. Probab. 30 181-196.
  • GRIFFIN, P. S. and MCCONNELL, T. R. (1992). On the position of a random walk at the time of first exit from a sphere. Ann. Probab. 20 825-854.
  • GRIFFIN, P. S. and MCCONNELL, T. R. (1995). Lp-boundedness of the overshoot in multidimensional renewal theory. Ann. Probab. 23 2022-2056.
  • KESTEN, H. (1969). Hitting probabilities of single points for processes with stationary independent increments. Mem. Amer. Math. Soc. 93.
  • KESTEN, H. (1970). The limit points of a normalized random walk. Ann. Math. Statist. 41 1173- 1205.
  • KESTEN, H. and MALLER, R. A. (1998). Random walks crossing power law boundaries. Studia Sci. Math. Hungarica 34 219-252.
  • MILLAR, P. R. (1973). Exit properties of stochastic processes with stationary independent increments. Trans. Amer. Math. Soc. 178 459-479.
  • PRUITT, W. E. (1981). The growth of random walks and Lévy processes. Ann. Probab. 9 948-956.
  • ROGOZIN, B. A. (1966). On distributions of functionals related to boundary crossing problems for processes with independent increments. Theory Probab. Appl. 11 580-591.
  • SPITZER, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.