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. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a partial sum process weakly converges in a function space if and only if the corresponding process defined through sums of independent random variables weakly converges. As a consequence of this result, necessary and sufficient conditions for weak convergence to a stable process are established. A counterexample showing that these conditions are not necessary for the one-dimensional convergence is presented. Very few results on the necessity part are known in the literature.
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REFERENCES
R. Arratia, A. D. Barbour, and S. Tavaré, Poisson process approximations for the Ewens sampling formula, Ann. Applied Probab., 2, 519–535 (1992).
R. Arratia and S. Tavaré, Limit theorems for combinatorial structures via discrete process approximations, Random Structures and Algorithms, 3, 321–345 (1992).
G. J. Babu, A note on the invariance principle for additive functions, Sankhy?A, 35, 307–310 (1973).
G. J. Babu and E. Manstavi?ius, Brownian motion for random permutations, Sankhy?A, 61, 312–327 (1999).
G. J. Babu and E. Manstavi?ius, Random permutations and the Ewens sampling formula in genetics, in: Probab. Theory and Math. Statist., Proc. 7th Vilnius Conf. (1998), B. Grigelionis et al. (Eds.), VSP, Utrecht/TEV, Vilnius (1999), pp. 33–42.
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).
J. M. DeLaurentis and B. G. Pittel, Random permutations and the Brownian motion, Pacific J. Math., 119, 287–301 (1985).
P. Donnelly, T. G. Kurtz, and S. Tavaré, On the functional central limit theorem for the Ewens sampling formula, Ann. Appl. Probab., 1, 539–545 (1991).
W. J. Ewens, The sampling theory of selectively neutral alleles, Theoret. Pop. Biol., 3, 87–112 (1972).
B. M. Hambly, P. Keevash, N. O'Connell, and D. Stark, The characteristic polynomial of a random permutation matrix, Stoch. Proc. Appl., 90, 335–346 (2000).
J. C. Hansen, A functional central limit theorem for the Ewens sampling formula, J. Appl. Probab., 27, 28–43 (1990).
J. Kubilius, Probabilistic Methods in the Theory of Numbers, Amer. Math. Soc., Providence, RI (1964).
E. Manstavi?ius, Arithmetic simulation of stochastic processes, Lith. Math. J., 24(3), 276–285 (1984).
E. Manstavi?ius, Additive functions and stochastic processes, Lith. Math. J., 24, 52–61 (1985).
E. Manstavi?ius, Additive and multiplicative functions on random permutations, Lith. Math. J., 36, 276–285 (1996).
V. V. Petrov, Sums of Independent Random Variables, Springer, New York (1975).
K. Wieand, Eigenvalue Distributions of Random Matrices in the Permutation Group and Compact Lie Groups, PhD thesis, Harvard University (1998).
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Babu, G.J., Manstavičius, E. Infinitely Divisible Limit Processes for the Ewens Sampling Formula. Lithuanian Mathematical Journal 42, 232–242 (2002). https://doi.org/10.1023/A:1020265607917
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DOI: https://doi.org/10.1023/A:1020265607917