Source: Ann. Appl. Probab. Volume 8, Number 4
(1998), 975-994.
Large deviation principles are established for some random
combinatorial structures including the Ewens sampling formula and the Pitman
sampling formula. A path-level large deviation principle is established for the
former on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence
topology, and the rate function is the same as for a Poisson process justifying
the Poisson process approximation for the Ewens sampling formula at the large
deviation level. A large deviation principle for the total number of parts in a
partition is obtained for the Pitman formula; here the rate function depends
only on one of the two parameters which display the different roles of the two
parameters at different scales. In addition to these large deviation results,
we also provide an embedding scheme which gives the Pitman sampling formula. A
product of this embedding is an intuitive alternate proof of a result of Pitman
on the limiting total number of parts.
References
[1] Antoniak, C. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2 1152-1174.
[2] Arratia, R., Barbour, A. D. and Tavar´e, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519-535.
[3] DeLaurentis, J. M. and Pittel, B. G. (1985). Random permutations and Brownian motion. Pacific J. Math. 119 287-301.
[4] Dembo, A. and Zeitouni, O. (1993). Large Deviations and Applications. Jones and Bartlett, Boston.
[5] Donnelly, P., Kurtz, T. G. and Tavar´e, S. (1991). On the functional central limit theorem for the Ewens sampling formula. Ann. Appl. Probab. 1 539-545.
[6] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biol. 3 87-112.
[7] Feng, S. and Hoppe, F. M. (1996). Models for partition structures. Unpublished manuscript.
[8] Goncharov, V. L. (1944). Some facts from combinatorics. Izv. Akad. Nauk SSSR Ser. Mat. 8 3-48.
[9] Hansen, J. C. (1990). A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27 28-43.
[10] Hoppe, F. M. (1984). P´oly a-like urns and the Ewens sampling formula. J. Math. Biol. 20 91-94.
[11] Karlin, S. and McGregor, J. (1967). The number of mutant forms maintained in a population. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 415-438. Univ. California Press, Berkeley.
Mathematical Reviews (MathSciNet):
MR214154
[12] Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. Roy. Soc. London Ser. A 361 1-20.
[13] Ly nch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Probab. 15 610-627.
[14] Mogulskii, A. A. (1993). Large deviations for processes with independent increments. Ann. Probab. 21 202-215.
[15] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158.
[16] Pitman, J. (1996). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79-96.
[17] Pitman, J. (1997). Notes on the two parameter generalization of the Ewens random partition structure. Unpublished manuscript.
[18] Shepp, L. A. and Lloy d, S. P. (1966). Ordered cy cle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340-357.
[19] Tavar´e, S. (1987). The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25 161-168.
