Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T00:50:18.486Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent

Part of: Graph theory Combinatorial probability Markov processes

Published online by Cambridge University Press:  30 March 2016

Martin Möhle
Martin Möhle*
Affiliation:
Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = limn→∞h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.

Keywords

Asymptotic hitting probabilityBolthausen-Sznitman coalescentgenerating function

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60C05: Combinatorial probability
Secondary: 05C05: Trees 92D15: Problems related to evolution
Type
Part 3. Biological applications
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 87 - 97
Copyright
Copyright © Applied Probability Trust 2014 

References

Aliprantis, C. D., and Tourky, R. (2007). Cones and Duality (Graduate Studies Math. 84). American Mathematical Society, Providence, RI.Google Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Bolthausen, E., and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.CrossRefGoogle Scholar
Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2007). Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. Stoch. Process. Appl. 117, 1404–1421.CrossRefGoogle Scholar
Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34, 319–336.Google Scholar
Flajolet, P., and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.Google Scholar
Gnedin, A. (2004). The Bernoulli sieve. Bernoulli 10, 79–96.CrossRefGoogle Scholar
Gnedin, A., Iksanov, A., and Marynych, A. (2011). On Λ-coalescents with dust component. J. Appl. Prob. 48, 1133–1151.Google Scholar
Meir, A., and Moon, J. W. (1970). Cutting down random trees. J. Austral. Math. Soc. 11, 313–324.Google Scholar
Meir, A., and Moon, J. W. (1974). Cutting down recursive trees. Math. Biosci. 21, 173–181.CrossRefGoogle Scholar
Möhle, M. (2005). Convergence results for compound Poisson distributions and applications to the standard Luria-Delbrück distribution. J. Appl. Prob. 42, 620–631.CrossRefGoogle Scholar
Norris, J. R. (1997). Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
Panholzer, A. (2004). Destruction of recursive trees. In Mathematics and Computer Science III, Birkhäuser, Basel, pp. 267–280.CrossRefGoogle Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.Google Scholar
Roman, S. (1984). The Umbral Calculus. Academic Press.Google Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.CrossRefGoogle Scholar