The distribution of the spine of a Fleming-Viot type process. (English) Zbl 1401.60060
Summary: We show uniqueness of the spine of a Fleming-Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number of particles goes to infinity, the distribution of the spine converges to that of the driving process conditioned to stay alive forever, the branching rate for the spine is twice that of a generic particle in the system, and every side branch has the distribution of the unconditioned generic branching tree.
MSC:
| 60G17 | Sample path properties |
References:
| [1] | Athreya, K. B.; Ney, P. E., (Branching Processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, (1972), Springer-Verlag New York-Heidelberg) · Zbl 0259.60002 |
| [2] | Bass, R. F., The measurability of hitting times, Electron. Commun. Probab., 15, 99-105, (2010) · Zbl 1202.60054 |
| [3] | Bieniek, M.; Burdzy, K.; Finch, S., Non-extinction of a Fleming-Viot particle model, Probab. Theory Related Fields, 153, 1-2, 293-332, (2012) · Zbl 1253.60089 |
| [4] | Bieniek, M.; Burdzy, K.; Pal, S., Extinction of Fleming-Viot-type particle systems with strong drift, Electron. J. Probab., 17, 11, 1-15, (2012) · Zbl 1258.60031 |
| [5] | Billingsley, P., (Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, (1999), John Wiley & Sons, Inc. New York) · Zbl 0944.60003 |
| [6] | Brémaud, P., Point processes and queues, (Martingale Dynamics, Springer Series in Statistics, (1981), Springer-Verlag New York-Berlin) · Zbl 0478.60004 |
| [7] | Burdzy, K.; Hołyst, R.; March, P., A Fleming-Viot particle representation of the Dirichlet Laplacian, Comm. Math. Phys., 214, 3, 679-703, (2000) · Zbl 0982.60078 |
| [8] | Collet, P.; Martínez, S.; Martín, J. San, Quasi-stationary distributions, (Markov Chains, Diffusions and Dynamical Systems, Probability and its Applications (New York), (2013), Springer Heidelberg) · Zbl 1261.60002 |
| [9] | Darroch, J. N.; Seneta, E., On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Probab., 4, 192-196, (1967) · Zbl 0168.16303 |
| [10] | Dawson, D. A.; Perkins, E. A., Historical processes, Mem. Amer. Math. Soc., 93, 454, (1991) · Zbl 0754.60062 |
| [11] | Depperschmidt, A.; Greven, A.; Pfaffelhuber, P., Tree-valued Fleming-Viot dynamics with mutation and selection, Ann. Appl. Probab., 22, 6, 2560-2615, (2012) · Zbl 1316.92048 |
| [12] | Donnelly, P.; Kurtz, T. G., A countable representation of the Fleming-Viot measure-valued diffusion, Ann. Probab., 24, 2, 698-742, (1996) · Zbl 0869.60074 |
| [13] | Dudley, R. M., Distances of probability measures and random variables, Ann. Math. Statist., 39, 1563-1572, (1968) · Zbl 0169.20602 |
| [14] | Engländer, J.; Kyprianou, A. E., Local extinction versus local exponential growth for spatial branching processes, Ann. Probab., 32, 1A, 78-99, (2004) · Zbl 1056.60083 |
| [15] | Etheridge, A. M., (An Introduction to Superprocesses, University Lecture Series, vol. 20, (2000), American Mathematical Society Providence, RI) · Zbl 0971.60053 |
| [16] | Evans, S. N., Two representations of a conditioned superprocess, Proc. Roy. Soc. Edinburgh Sect. A, 123, 5, 959-971, (1993) · Zbl 0784.60052 |
| [17] | Greven, A.; Limic, V.; Winter, A., Representation theorems for interacting Moran models, interacting Fisher-wright diffusions and applications, Electron. J. Probab., 10, 39, 1286-1356, (2005), (electronic) · Zbl 1109.60082 |
| [18] | Greven, A.; Pfaffelhuber, P.; Winter, A., Tree-valued resampling dynamics martingale problems and applications, Probab. Theory Related Fields, 155, 3-4, 789-838, (2013) · Zbl 1379.60099 |
| [19] | Grigorescu, I.; Kang, M., Immortal particle for a catalytic branching process, Probab. Theory Related Fields, 153, 1-2, 333-361, (2012) · Zbl 1251.60064 |
| [20] | Hénard, O., Change of measure in the lookdown particle system, Stoch. Process. Appl., 123, 6, 2054-2083, (2013) · Zbl 1302.60107 |
| [21] | Jagers, P., Branching Processes with Biological Applications, (1975), Wiley-Interscience [John Wiley & Sons] London-New York-Sydney · Zbl 0356.60039 |
| [22] | Lyons, R.; Pemantle, R.; Peres, Y., Conceptual proofs of \(L \log L\) criteria for mean behavior of branching processes, Ann. Probab., 23, 3, 1125-1138, (1995) · Zbl 0840.60077 |
| [23] | Moran, P. A.P., Random processes in genetics, Proc. Cambridge Philos. Soc., 54, 60-71, (1958) · Zbl 0091.15701 |
| [24] | Seidel, P., The historical process of the spatial moran model with selection and mutation, (2014), Friedrich-Alexander-Universität Erlangen-Nürnberg, (Ph.D. thesis) |
| [25] | Villemonais, D., General approximation method for the distribution of Markov processes conditioned not to be killed, ESAIM Probab. Stat., 18, 441-467, (2014) · Zbl 1310.82032 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
