Cramér’s estimate for Lévy processes.

*(English)*Zbl 0809.60085One of the most useful results in applied probability is Cramér’s estimate, which gives the asymptotic behaviour of the tail of the distribution of the all-time maximum of a random walk with negative drift. This estimate is an immediate consequence of the renewal theorem applied to the renewal process formed by the ascending ladder heights in the associated random walk. The authors show that exactly the same method works for Lévy processes, which, being processes with stationary and independent increments, are the continuous time analogues of random walks. The only complication is that the set of increasing ladder heights is not usually discrete, but coincides with the range of a certain subordinator. However, the authors observe that the potential function of this subordinator coincides with the renewal function of a certain renewal process, so that they can still appeal to the renewal theorem.

Reviewer: Z.Rychlik (Lublin)

##### MSC:

60J99 | Markov processes |

##### Keywords:

Lévy processes; Cramér’s estimate; random walk with negative drift; renewal process; renewal theorem
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\textit{J. Bertoin} and \textit{R. A. Doney}, Stat. Probab. Lett. 21, No. 5, 363--365 (1994; Zbl 0809.60085)

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##### References:

[1] | Bingham, N.H., Fluctuation theory in continuous time, Adv. appl. probab., 7, 705-766, (1975) · Zbl 0322.60068 |

[2] | Blumenthal, R.M.; Getoor, R.K., Markov processes and potential theory, (1968), Academic Press New York · Zbl 0169.49204 |

[3] | Doney, R.A., Hitting probabilities for spectrally positive Lévy processes, J. London math. soc. (2), 44, 566-576, (1991) · Zbl 0699.60061 |

[4] | Feller, W., An introduction to probability theory and its applications, II, (1966), Wiley New York · Zbl 0138.10207 |

[5] | Fristedt, B., Sample functions of processes with stationary independent increments, Adv. probab., 3, 241-396, (1973) |

[6] | Iglehart, D.L., Extreme values for the GI/G/1 queue, Ann. math. statist., 43, 627-635, (1972) · Zbl 0238.60072 |

[7] | Port, S.C.; Stone, C.J., Infinitely divisible processes and their potential theory, I, Ann. inst. Fourier (Grenoble), 21, 157-275, (1971) · Zbl 0195.47601 |

[8] | Williams, D., Diffusions, Markov processes, and martingales, I, (1979), Wiley New York |

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