Bernoulli

  • Bernoulli
  • Volume 17, Number 4 (2011), 1217-1247.

Self-similar scaling limits of non-increasing Markov chains

Bénédicte Haas and Grégory Miermont

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Abstract

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately rescaled, converges in distribution, as n → ∞, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0.

We discuss various applications to the study of random walks with a barrier, of the number of collisions in Λ-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in our paper, Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees (2010).

Article information

Source
Bernoulli Volume 17, Number 4 (2011), 1217-1247.

Dates
First available in Project Euclid: 4 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1320417502

Digital Object Identifier
doi:10.3150/10-BEJ312

Mathematical Reviews number (MathSciNet)
MR2854770

Zentralblatt MATH identifier
1263.92034

Keywords
absorption time Λ-coalescents random walks with a barrier regenerative compositions regular variation self-similar Markov processes

Citation

Haas, Bénédicte; Miermont, Grégory. Self-similar scaling limits of non-increasing Markov chains. Bernoulli 17 (2011), no. 4, 1217--1247. doi:10.3150/10-BEJ312. https://projecteuclid.org/euclid.bj/1320417502


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References

  • [1] Bertoin, J. and Yor, M. (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Comm. Probab. 6 95–106 (electronic).
  • [2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley.
  • [3] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989). Regular Variation. Encyclopedia of Mathematics and its Applications 27 Cambridge: Cambridge Univ. Press.
  • [4] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion. Bibl. Rev. Mat. Iberoamericana 73–130. Madrid: Rev. Mat. Iberoamericana.
  • [5] Delmas, J.-F., Dhersin, J.-S. and Siri-Jegousse, A. (2008). Asymptotic results on the length of coalescent trees. Ann. Appl. Probab. 18 997–1025.
  • [6] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes. New York: Wiley.
  • [7] Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Probab. 45 1186–1195.
  • [8] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479.
  • [9] Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 468–492.
  • [10] Gnedin, A. and Yakubovich, Y. (2007). On the number of collisions in Λ-coalescents. Electron. J. Probab. 12 1547–1567 (electronic).
  • [11] Haas, B. and Miermont, G. (2010). Scaling limits of Markov branching trees, with applications to Galton–Watson and random unordered trees. Available at ArXiv:1003.3632.
  • [12] Haas, B., Pitman, J. and Winkel, M. (2009). Spinal partitions and invariance under re-rooting of continuum random trees. Ann. Probab. 37 1381–1411.
  • [13] Iksanov, A., Marynych, A. and Möhle, M. (2009). On the number of collisions in beta (2, b)-coalescents. Bernoulli. 15 829–845.
  • [14] Iksanov, A. and Möhle, M. (2008). On the number of jumps of random walks with a barrier. Adv. in Appl. Probab. 40 206–228.
  • [15] Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62–78.
  • [16] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205–225.
  • [17] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [18] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
  • [19] Schweinsberg, J. (2000). A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Comm. Probab. 5 1–11 (electronic).
  • [20] Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638–660.