Abstract
The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ⋂ [0, 1] for a self-similar random set S ⊂ ℝ+ are those that are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of n is defined by the first n terms of a random binary sequence of infinite length. The locations of 1’s in the sequence are the positions visited by an increasing time-homogeneous Markov chain on the positive integers if and only if S = exp(−W) for some stationary regenerative random subset W of the real line. Complementing our study presented in previous papers, we identify self-similar Markov composition structures associated with the two-parameter family of partition structures. Bibliography: 19 titles.
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References
R. Arratia, “On the central role of scale invariant Poisson processes on (0, ∞),” in: Microsurveys in Discrete Probability (1997), pp. 21–41.
R. Arratia, A. D. Barbour, and S. Tavaré, Logarithmic Combinatorial Structures: a Probabilistic Approach, EMS Monographs in Mathematics, European Mathematical Society, Zürich (2003).
A. D. Barbour and A. V. Gnedin, “Regenerative compositions in the case of slow variation” (2005), arXiv:math.PR/0505171.
J. Bunge and C. Goldie, “Record sequences and their applications,” in: Handbook of Statistics, 19, D. N. Shanbhag and C. R. Rao (eds.), Elsevier (2001), pp. 277–308.
P. Donnelly and P. Joyce, “Consistent ordered sampling distributions: characterization and convergence,” Adv. Appl. Probab., 23, 229–258 (1991).
A. V. Gnedin, “The representation of composition structures,” Ann. Probab., 25, 1437–1450 (1997).
A. V. Gnedin, “The Bernoulli sieve,” Bernoulli, 10, 79–96 (2004).
A. V. Gnedin, “Three sampling formulas,” Combin. Probab. Comput., 13, 185–193 (2004).
A. V. Gnedin and J. Pitman, “Regenerative composition structures,” Ann. Probab., 33, 445–479 (2005).
A. V. Gnedin and J. Pitman, “Regenerative partition structures,” Electron. J. Combin., 11, No. 2, R12 (2004/2005).
A. V. Gnedin, J. Pitman, and M. Yor, “Asymptotic laws for compositions derived from transformed subordinators,” Ann. Probab., 34, 468–492 (2006).
A. V. Gnedin, J. Pitman, and M. Yor, “Asymptotic laws for regenerative compositions: gamma subordinators and the like” (2004); arXiv:math.PR/0405440.
S. Nacu, “Increments of random partitions,” Combin. Probab. Comput., 15, 589–595 (2006).
J. Pitman, “Partition structures derived from Brownian motion and stable subordinators,” Bernoulli, 3, 79–96 (1997).
J. Pitman, Combinatorial Stochastic Processes, Lect. Notes St. Flour Course (July 2002), to appear in Lect. Notes Math. Available via www.stat.berkeley.edu.
J. Pitman and M. Yor, “The two-parameter Poisson-Dirichlet distribution derived from transformed subordinator,” Ann. Probab., 25, 855–900 (1996).
J. Pitman and M. Yor, “Random discrete distributions derived from self-similar random sets,” Electron. J. Probab., 1, 1–28 (1996).
M. I. Taksar, “Stationary Markov sets,” Lect. Notes Math., 1247, 302–340 (1986).
J. E. Young, “Partition-valued stochastic processes with applications,” PhD thesis, University of California at Berkeley (1995).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 59–84.
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Gnedin, A., Pitman, J. Self-similar and Markov composition structures. J Math Sci 140, 376–390 (2007). https://doi.org/10.1007/s10958-007-0447-0
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DOI: https://doi.org/10.1007/s10958-007-0447-0