Abstract
The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arratia, R. and Tavaré, S. (1992). Limit theorems for combinatorial structures via discrete process approximations, Random Structures Algorithms, 3, 321–345.
Arratia, R., Barbour, A. D. and Tavaré, S. (1992). Poisson process approximations for the Ewens sampling formula, Ann. Appl. Probab., 2, 519–535.
Babu, G. J. (1973). A note on the invariance principle for additive functions, Sankhyā Ser. A, 35, 307–310.
Babu, G. J. and Manstavičius, E. (1999). Brownian motion for random permutations, Sankhyā Ser. A, 61, 312–327.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular variation, 2nd ed., Cambridge University Press, Cambridge, Massachusetts.
DeLaurentis, J. M. and Pittel, B. G. (1985). Random permutations and the Brownian motion, Pacific J. Math., 119, 287–301.
Donnelly, P., Kurtz, T. G. and Tavaré, S. (1991). On the functional central limit theorem for the Ewens Sampling Formula, Ann. Appl. Probab., 1, 539–545.
Drmota, M. and Tichy, R. F. (1997). Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, 1651, Springer, Berlin.
Erdös, P. and Turán, P. (1965). On some problems of a statistical group theory I, Z. Wahrsch. Verw. Gebiete, 4, 175–186.
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles, Theoretical Population Biology, 3, 87–112.
Ewens, W. J. (1979). Mathematical Population Genetics, Springer, Berlin.
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume 1, 2nd ed., Wiley, New York.
Hansen, J. C. (1990). A functional central limit theorem for the Ewens Sampling Formula, J. Appl. Probab., 27, 28–43.
Hirth, U. M. (1997a). A Poisson approximation for the Dirichlet distribution, the Ewens sampling formula and the Griffith-Engen-McCloskey law by the Stein-Chen coupling method, Bernoulli, 3, 225–232.
Hirth, U. M. (1997b). Probabilistic number theory, the GEM/Poisson-Dirichlet distribution and the arc-sine law, Combin. Probab. Comput., 6, 57–77.
Hirth, U. M. (1997c). From GEM back to Dirichlet via Hoppe's urn, Combin. Probab. Comput., 6, 185–195.
Kingman, J. F. C. (1980). Mathematics of Genetic Diversity, SIAM, Philadelphia, Pennsylvania.
Kubilius, J. (1964). Probabilistic Methods in the Theory of Numbers, Translations of Mathematical Monographs, 11, AMS, Providence, Rhode Island.
Manstavičius, E. (1984). Arithmetic simulation of stochastic processes, Lithuanian Math. J., 24, 276–285.
Manstavičius, E. (1985). Additive functions and stochastic processes, Lithuanian Math. J., 25, 52–61.
Manstavičius, E. (1996), Additive and multiplicative functions on random permutations, Lithuanian Math. J., 36(4), 400–408.
Petrov, V. V. (1975). Sums of Independent Random Variables, Springer, New York.
Philipp, W. (1973). Arithmetic functions and Brownian motion, Proc. Sympos. Pure Math., 24, 233–246.
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.
Tenenbaum, G. (1995). Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge.
Timofeev, N. M. and Usmanov, H. H. (1984). Arithmetic modelling of random processes with independent increments, Dokl. Akad. Nauk Respub. Tadzhikistan, 27(10), 556–559 (Russian).
Timofeev, N. M. and Usmanov, H. H. (1986). On a class of arithmetic models of random processes, Dokl. Akad. Nauk Respub. Tadzhikistan, 29(6), 330–334 (Russian).
Author information
Authors and Affiliations
About this article
Cite this article
Babu, G.J., Manstavičius, E. Limit Processes with Independent Increments for the Ewens Sampling Formula. Annals of the Institute of Statistical Mathematics 54, 607–620 (2002). https://doi.org/10.1023/A:1022419328971
Issue Date:
DOI: https://doi.org/10.1023/A:1022419328971