Lévy processes conditioned on having a large height process
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, p. 982-1013

In the present work, we consider spectrally positive Lévy processes (X t ,t0) not drifting to + and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with X) before hitting 0. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law x of this conditioned process (starting at x>0) is defined as a Doob h-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is (ρ ˜ t (dz)e αz +I t )1 {tT 0 } for some α and where (I t ,t0) is the past infimum process of X, where (ρ ˜ t ,t0) is the so-called exploration process defined in [10] and where T 0 is the hitting time of 0 for X. Under x , we also obtain a path decomposition of X at its minimum, which enables us to prove the convergence of x as x0. When the process X is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of X. The computations are easier in this case because X can be viewed as the contour process of a (sub)critical splitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

Dans ce travail, on considère des processus de Lévy (X t ,t0) ne dérivant pas vers + et on s’intéresse à leur conditionnement à atteindre des hauteurs arbitrairement grandes (au sens du processus des hauteurs associé à X) avant de toucher 0. On obtient ainsi une nouvelle manière de conditionner des processus de Lévy à rester positifs. La loi (honnête) x de ce processus conditionné (partant de x>0) est définie selon une h-transformée de Doob à l’aide d’une martingale. En ce qui concerne les processus de Lévy ayant des trajectoires à variation infinie, cette martingale est (ρ ˜ t (dz)e αz +I t )1 {tT 0 } pour un certain α et où (I t ,t0) est le processus infimum de X, où (ρ ˜ t ,t0) est le processus d'exploration défini dans [10] et où T 0 est le temps d’atteinte de 0 par X. Sous x , on obtient également une décomposition de la trajectoire de X en son minimum; ce qui permet de prouver la convergence de x quand x0. Lorsque le processus X est un processus de Poisson composé compensé, la martingale est définie à partir des sauts du processus infimum futur de X. Les preuves sont plus simples dans ce cas puisque on peut voir X comme le processus de contour d’un arbre de ramification (sous)critique. Dans ce cas, on énonce aussi une caractérisation alternative du processus conditionné dans l'esprit des décompositions spinales.

DOI : https://doi.org/10.1214/12-AIHP491
Classification:  60G51,  60J80,  60J85,  60G44,  60K25,  60G07,  60G57
Keywords: Lévy process, height process, Doob harmonic transform, splitting tree, spine decomposition, Size-biased distribution, queueing theory
@article{AIHPB_2013__49_4_982_0,
     author = {Richard, Mathieu},
     title = {L\'evy processes conditioned on having a large height process},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {4},
     year = {2013},
     pages = {982-1013},
     doi = {10.1214/12-AIHP491},
     mrnumber = {3127910},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_4_982_0}
}
Richard, Mathieu. Lévy processes conditioned on having a large height process. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, pp. 982-1013. doi : 10.1214/12-AIHP491. http://www.numdam.org/item/AIHPB_2013__49_4_982_0/

[1] R. Abraham and J.-F. Delmas. Feller property and infinitesimal generator of the exploration process. J. Theoret. Probab. 20 (2007) 355-370. | MR 2324536

[2] D. Aldous. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 (1991) 228-266. | MR 1102319

[3] K. B. Athreya and P. E. Ney. Branching Processes. Grundlehren der mathematischen Wissenschaften 196. Springer, New York, 1972. | MR 373040

[4] J. Azéma and M. Yor. Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) 90-115. Lecture Notes in Math. 721. Springer, Berlin, 1979. | Numdam | MR 544782

[5] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge, 1996. | MR 1406564

[6] L. Chaumont. Sur certains processus de Lévy conditionnés à rester positifs. Stochastics Stochastics Rep. 47 (1994) 1-20. | MR 1787140

[7] L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. | MR 1419491

[8] L. Chaumont and R. A. Doney. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005) 948-961 (electronic). | MR 2164035

[9] T. Duquesne. Continuum random trees and branching processes with immigration. Stochastic Process. Appl. 119 (2009) 99-129. | MR 2485021

[10] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi+147. | MR 1954248

[11] J. Geiger. Size-biased and conditioned random splitting trees. Stochastic Process. Appl. 65 (1996) 187-207. | MR 1425355

[12] J. Geiger. Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab. 36 (1999) 301-309. | MR 1724856

[13] J. Geiger and G. Kersting. Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 111-126. IMA Vol. Math. Appl. 84. Springer, New York, 1997. | MR 1601713

[14] K. Hirano. Lévy processes with negative drift conditioned to stay positive. Tokyo J. Math. 24 (2001) 291-308. | MR 1844435

[15] D. P. Kennedy. Some martingales related to cumulative sum tests and single-server queues. Stochastic Processes Appl. 4 (1976) 261-269. | MR 420834

[16] A. Lambert. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 (2002) 42-70. | MR 1883717

[17] A. Lambert. Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 (2007) 420-446. | MR 2299923

[18] A. Lambert. Population dynamics and random genealogies. Stoch. Models 24 (2008) 45-163. | MR 2466449

[19] A. Lambert. The contour of splitting trees is a Lévy process. Ann. Probab. 38 (2010) 348-395. | MR 2599603

[20] J.-F. Le Gall and Y. Le Jan. Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 (1998) 213-252. | MR 1617047

[21] Z.-H. Li. Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68 (2000) 68-84. | MR 1727226

[22] V. Limic. A LIFO queue in heavy traffic. Ann. Appl. Probab. 11 (2001) 301-331. | MR 1843048

[23] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of LlogL criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125-1138. | MR 1349164

[24] P. Millar. Zero-one laws and the minimum of a Markov process. Trans. Am. Math. Soc. 226 (1977) 365-391. | MR 433606

[25] L. Nguyen-Ngoc. Limiting laws and penalization of certain Lévy processes by a function of their maximum. Teor. Veroyatn. Primen. 55 (2010) 530-547. | MR 2768536

[26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der mathematischen Wissenschaften 293. Springer, Berlin, 1999. | MR 1725357

[27] P. Robert. Stochastic Networks and Queues, french edition. Applications of Mathematics (New York) 52. Stochastic Modelling and Applied Probability. Springer, Berlin, 2003. | MR 1996883

[28] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795-798. | MR 22045