A hitting time for Lévy processes, with application to dams and branching processes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 5 (1996) no. 3, pp. 521-544. 
@article{AFST_1996_6_5_3_521_0,
     author = {Pakes, Anthony G.},
     title = {A hitting time for {L\'evy} processes, with application to dams and branching processes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {521--544},
     publisher = {Universit\'e Paul Sabatier},
     address = {Toulouse},
     volume = {Ser. 6, 5},
     number = {3},
     year = {1996},
     mrnumber = {1440948},
     zbl = {0879.60074},
     language = {en},
     url = {http://www.numdam.org/item/AFST_1996_6_5_3_521_0/}
}
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Pakes, Anthony G. A hitting time for Lévy processes, with application to dams and branching processes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 5 (1996) no. 3, pp. 521-544. http://www.numdam.org/item/AFST_1996_6_5_3_521_0/

[1] Bingham (N.H.) .- Fluctuation theory in continuous time, Adv. Appl. Prob. 7 (1975), pp. 705-766. | MR | Zbl

[2] Bingham (N.H.) .- Continuous branching processes and spectral positivity, Stoch. Processes Appl. 4 (1976), pp. 217-242. | MR | Zbl

[3] Bingham (N.H.) .- The work of Lajos Takács on probability theory. J. Appl. Prob. 31A (1994), pp. 29-39. | MR | Zbl

[4] Bingham (N.H.), Goldie (C.M.) and Teugels (J.F.) .- Regular Variation, C.U.P., Cambridge (1987). | MR | Zbl

[5] Bondesson (L.) . - Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statistics, Springer-Verlag, New York, 76 (1992). | MR | Zbl

[6] Borovkov (A.A.) .- On the first passage time for one class of processes with independent increments, Theor. Prob. Appl. 10 (1965), pp. 331-334. | MR | Zbl

[7] Borovkov (A.A.) .- Stochastic Processes in Queueing Theory, Springer-Verlag, New York (1976). | MR | Zbl

[8] Consul (P.C.) and Shenton (L.R.) .- Some interesting properties of Lagrangian distributions, Comm. Statist. 2 (1973), pp. 263-272. | MR | Zbl

[9] Devroye (L.) .- A note on Linnik's distribution, Statist. Prob. Lett. 9 (1990), pp. 305-306. | MR | Zbl

[10] Devroye (L.) .- The branching process method in Lagrange random variate generation, Comm. Statist. Simula. 21 (1992), pp. 1-14. | Zbl

[11] Feller (W.) . - Probability Theory and its Applications, Wiley, New York, 2nd ed., 2 (1971).

[12] Fristedt (B.) .- Sample functions of stochastic processes with stationary, independent increments, In: P. E. Ney and S. Port eds, Advances in Probability, Dekker, New York, 5 (1974), pp. 241-396. | MR | Zbl

[13] Gani (J.) and Prabhu (N.U.) .- A storage model with continuous infinitely divisible inputs, Proc. Camb. Phil. Soc. 59 (1963), pp. 417-429. | MR | Zbl

[14] Gihman (I.I.) and Skorohod (A.V.) .- The Theory of Stochastic Processes II, Springer-Verlag, Berlin (1975). | MR | Zbl

[15] Harrison (J.M.) .- The supremum distribution of a Lévy process with no negative jumps, Adv. Appl. Prob. 9 (1977), pp. 417-422. | MR | Zbl

[16] Hasofer (A.M.) .- On the distribution of the time to first emptiness of a store with stochastic input, J. Aust. Math. Soc. 4 (1964), pp. 506-517. | MR | Zbl

[17] Hougaard (P.) . - Survival models for heterogeneous populations derived from stable distributions, Biometrika 73 (1986), pp. 387-396. | MR | Zbl

[18] Ibragimov (I.A.) and Linnik Yu (V.) .- Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen (1971). | MR | Zbl

[19] Johnson (N.L.), Kotz (S.) and Kemp (A.W.) .- Univariate Discrete Distributions, Wiley, New York, 2nd ed. (1993). | MR | Zbl

[20] Kallenberg (P.J.M.) .- Branching Processes with Continuous State Space, Math. Centrum, Amsterdam (1979). | MR | Zbl

[21] Keilson (J.) . - The first passage time density for homogeneous skipfree walks on the continuum, Ann. Math. Statist. 34 (1963), pp. 1003-1011. | MR | Zbl

[22] Kendall (D.G.) .- Some problems in the theory of dams, J. Roy. Statist. Soc. Ser. B. 19 (1957), pp. 207-212. | MR | Zbl

[23] Kingman (J.F.C.) .- On continuous time models in the theory of dams, J. Aust. Math. Soc. 3 (1963), pp. 480-487. | MR | Zbl

[24] Letac (G.) and Mora (M.) .- Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990), pp. 1-37. | MR | Zbl

[25] Moran (P.A.P.) .- An Introduction to Probability Theory, Clarendon Press, Oxford (1968). | MR | Zbl

[26] Otter (R.) .- The multiplicative process, Ann. Math. Statist. 20 (1949), pp. 206-224. | MR | Zbl

[27] Pakes (A.G.) .- Some limit theorems for continuous-state branching processes, J. Aust. Math. Soc. Ser. A. 44 (1988), pp. 71-87. | MR | Zbl

[28] Pakes (A.G.) and Speed (T.P.) .- Lagrange distributions and their limit theorems, SIAM J. Appl. Math. 32 (1977), pp. 745-754. | MR | Zbl

[29] Prabhu (N.U.) .- Stochastic Storage Processes, Springer-Verlag, New York (1980). | MR | Zbl

[30] Prabhu (N.U.) and Rubinovitch (M.) .- On a regenerative phenomenon occurring in a storage model, J. Roy. Statist. Soc. Ser. B. 32 (1970), pp. 354-361. | MR | Zbl

[31] Rogers (L.C.G.) . - The two-sided exit problem for spectrally positive Lévy processes, Adv. Appl. Prob. 22 (1990), pp. 486-487. | MR | Zbl

[32] Rosinski (J.) .- On a class of infinitely divisible processes represented as mixtures of Gaussian processes, In: S. Cambanis, G. Samarodnitsky et M. Taqqu, eds, Stable Processes and Related Topics, Birkäuser, Boston (1991), pp. 27-41. | MR | Zbl

[33] Seshadri (V.) .- Inverse-Gaussian Distributions: A Case Study in Natural Exponential Families, Clarendon Press, Oxford (1993). | MR

[34] Shtatland (E.S.) .- On local properties of processes with independent increments, Theor. Prob. Appl. 10 (1965), pp. 317-322. | Zbl

[35] Skorohod (A.V.) .- Random Processes with Independent Increments, Kluwer Academic Publishers, Dordrecht (1991). | MR | Zbl

[36] Stone (C.) .- Ratio limit theorems for random walks on groups, Trans. Amer. Math. Soc. 125 (1966), pp. 86-100. | MR | Zbl

[37] Takács (L.) . - The distribution of the content of a dam when the input process has stationary independent increments, J. Math. Mech. 15 (1966), pp. 101-112. | MR | Zbl

[38] Takács (L.) .- Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York (1967). | MR | Zbl

[39] Wendel (J.G.) .- Left-continuous random walk and the Lagrange expansion, Amer. Math. Monthly 82 (1975), pp. 494-499. | MR | Zbl

[40] Zolotarev (V.M.) .- A duality law in the class of infinitely divisible laws. English translation in Sel, Trans. Math. Statist. Prob. 5 (1961), pp. 201-209. | MR

[41] Zolotarev (V.M.) .- The first passage time of a level and the behavior at infinity for a class of processes with independent increments, Theor. Prob. Appl. 9 (1964), pp. 653-661. | MR | Zbl