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. MR51:2221 0335.60034 10.1214/aos/1176342871 euclid.aos/1176342871
[1] Antoniak, C. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2 1152-1174. MR51:2221 0335.60034 10.1214/aos/1176342871 euclid.aos/1176342871
[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. MR94a:60003 0756.60006 10.1214/aoap/1177005647 euclid.aoap/1177005647
[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. MR94a:60003 0756.60006 10.1214/aoap/1177005647 euclid.aoap/1177005647
[3] DeLaurentis, J. M. and Pittel, B. G. (1985). Random permutations and Brownian motion. Pacific J. Math. 119 287-301. MR86m:60027 0578.60033 euclid.pjm/1102706156
[3] DeLaurentis, J. M. and Pittel, B. G. (1985). Random permutations and Brownian motion. Pacific J. Math. 119 287-301. MR86m:60027 0578.60033 euclid.pjm/1102706156
[4] Dembo, A. and Zeitouni, O. (1993). Large Deviations and Applications. Jones and Bartlett, Boston. MR95a:60034[4] Dembo, A. and Zeitouni, O. (1993). Large Deviations and Applications. Jones and Bartlett, Boston. MR95a:60034
[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. MR93e:60064 0747.60013 10.1214/aoap/1177005837 euclid.aoap/1177005837
[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. MR93e:60064 0747.60013 10.1214/aoap/1177005837 euclid.aoap/1177005837
[6] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biol. 3 87-112. MR48:3526 10.1016/0040-5809(72)90035-4[6] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biol. 3 87-112. MR48:3526 10.1016/0040-5809(72)90035-4
[9] Hansen, J. C. (1990). A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27 28-43. 0704.92011 MR1039182 10.2307/3214593[9] Hansen, J. C. (1990). A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab. 27 28-43. 0704.92011 MR1039182 10.2307/3214593
[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. MR214154 0218.60074[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. MR214154 0218.60074
[12] Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. Roy. Soc. London Ser. A 361 1-20. MR58:26167 10.1098/rspa.1978.0089[12] Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. Roy. Soc. London Ser. A 361 1-20. MR58:26167 10.1098/rspa.1978.0089
[13] Ly nch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Probab. 15 610-627. MR88m:60076 0624.60045 10.1214/aop/1176992161 euclid.aop/1176992161
[13] Ly nch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Probab. 15 610-627. MR88m:60076 0624.60045 10.1214/aop/1176992161 euclid.aop/1176992161
[14] Mogulskii, A. A. (1993). Large deviations for processes with independent increments. Ann. Probab. 21 202-215. MR94g:60053 10.1214/aop/1176989401 euclid.aop/1176989401
[14] Mogulskii, A. A. (1993). Large deviations for processes with independent increments. Ann. Probab. 21 202-215. MR94g:60053 10.1214/aop/1176989401 euclid.aop/1176989401
[15] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. MR96e:60059 0821.60047 10.1007/BF01213386[15] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. MR96e:60059 0821.60047 10.1007/BF01213386
[16] Pitman, J. (1996). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79-96. MR1466546 10.2307/3318653 euclid.bj/1178291933
[16] Pitman, J. (1996). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79-96. MR1466546 10.2307/3318653 euclid.bj/1178291933
[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. MR33:3320 0156.18705 10.2307/1994483[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. MR33:3320 0156.18705 10.2307/1994483
[19] Tavar´e, S. (1987). The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25 161-168. MR89e:92042 0625.92010[19] Tavar´e, S. (1987). The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25 161-168. MR89e:92042 0625.92010
