Eternal additive coalescents and certain bridges with exchangeable increments. (English) Zbl 1019.60072

The author studies the additive coalescent processes, i.e., random processes which describe the evolution of the ranked sequence of masses in a system of clusters in which each pair of clusters, say with masses \(m_i\) and \(m_j,\) merges as a single cluster with mass \(m_i+m_j\) at rate \(\kappa(m_i,m_j)=m_i+m_j,\) independently of the other pairs. Aldous and Pitman have studied the asymptotic behavior of the additive coalescent processes. The author proposes a different and simpler approach to this problem based on partitions of the unit interval induced by certain bridges with exchangeable increments. The connection between additive coalescent and interval partitions is enlightened by an interpretation in terms of an aggregating server system.


60J25 Continuous-time Markov processes on general state spaces
60G09 Exchangeability for stochastic processes
Full Text: DOI


[1] Aldous, D. J. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812-854. · Zbl 0877.60010 · doi:10.1214/aop/1024404421
[2] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 3-48. · Zbl 0930.60096 · doi:10.2307/3318611
[3] Aldous, D. J. and Limic, V. (1998). The entrance boundary of the multiplicative coalescent. Electronic J. Probab. 3, paperno. 3. http://www.math.washington.edu/ ejpecp/ EjpVol3/paper3.abs.html. URL: · Zbl 0889.60080
[4] Aldous, D. J. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703- 1726. · Zbl 0936.60064 · doi:10.1214/aop/1022855879
[5] Aldous, D. J. and Pitman, J. (1999). Inhomogenous continuum random tree and the entrance boundary of the additive coalescent. Probab. Theory RelatedFields. · Zbl 0969.60015 · doi:10.1007/s004400000094
[6] Bertoin, J. (1993). Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 17-35. · Zbl 0786.60101 · doi:10.1016/0304-4149(93)90092-I
[7] Bertoin, J.(2000). Clustering statistics for sticky particles with Brownian initial velocity. J. Math. Pures Appl. 79 173-194. · Zbl 0959.60074 · doi:10.1016/S0021-7824(00)00147-1
[8] Bertoin, J. (2000). A fragmentation process connected to Brownian motion. Probab. Theory RelatedFields 117 289-301. · Zbl 0965.60072 · doi:10.1007/s004400000056
[9] Camarri, M. and Pitman, J. (2000). Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electronic J. Probab. 5. http://www.math.washington.edu/ ejpecp/ejp5contents.html. URL: · Zbl 0953.60030
[10] Chassaing, P. and Janson, S. (1999). A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab. · Zbl 1032.60076 · doi:10.1214/aop/1015345771
[11] Chassaing, P. and Louchard, G. (1999). Phase transition for parking blocks, Brownian excursions and coalescence. · Zbl 1032.60003
[12] Evans, S. N. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. H. Poincaré Probab. Statist. 34 339-383. · Zbl 0906.60058 · doi:10.1016/S0246-0203(98)80015-0
[13] Jacod, J. and Shiryaev, A. N. (1987). Limit theorems for stochastic processes. Springer, Berlin. · Zbl 0635.60021
[14] Kallenberg, O.(1973). Canonical representations and convergence criteria for processes with interchangeable increments.Wahrsch. Verw. Gebiete 27 23-36. · Zbl 0253.60060 · doi:10.1007/BF00736005
[15] Kallenberg, O. (1974). Path properties of processes with independent and interchangeable increments.Wahrsch. Verw. Gebiete 28 257-271. · Zbl 0266.60028 · doi:10.1007/BF00532944
[16] Knight, F. B. (1996). The uniform law for exchangeable and Lévy process bridges. Hommage a P. A. Meyer et J. Neveu. Astérisque 236 171-188. · Zbl 0867.60018
[17] Takács, L.(1966). Combinatorial methods in the theory of stochastic processes. Wiley, New York Vervaat, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 141-149. · Zbl 0392.60058 · doi:10.1214/aop/1176995155
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