Summary
LetX i,n′ i=1, ...n be a triangular array in the domain of attraction of a Lévy processX(t). Then the symmetric polynomial of orderk in theX i,n′
converges weakly to thek th order multiple integral with respect toX(t). Also, sums of powers of theX i,n converge jointly to the variations of the processX(t).
References
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Dynkin, E.B., Mandelbaum, A.: Symmetric statistics, Poisson point processes and multiple Wiener integrals. Ann. Stat.11, 739–745 (1983)
Denker, M., Grillenberger, Chr., Keller, G.: A note on invariance principles for von Mises' statistics. Metrika (to appear 1985.)
Feinsilver, P.J.: Special functions, probability semigroups and Hamiltonian flows. Lect. Notes in Math.696. Berlin-Heidelberg-New York: Springer 1978
Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Reading, Mass: Addison-Wesley 1954
Holley, R., Strock, D.W.: Central limit phenomena of various interacting systems. Ann. Math.110, 333–393 (1979)
Itô, K.: Stochastic Processes. Lecture Notes Series16, Mathematisk Institut, Aarhus Universitet (1969)
Mandelbaum, A., Taqqu, M.S.: Invariance principle for symmetric statistics. Ann. of Statistics12, 483–496 (1984)
Meyer, P.-A.: Un cours sur les intégrales stochastiques. Sem. Probl. X. Lect. Notes Math.511, 245–400. Berlin-Heidelberg-New York: Springer 1976
Rvaceva, E.L.: On domains of attraction of multi-dimensional distributions. Selected Transl. in Probab. v.2, 1962, 183–205 (1954)
Rubin, H., Vitale, R.A.: Asymptotic distribution of symmetric statistics. Ann. of Statistics8, 165–170 (1980)
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The second author is supported in part by NSF grant ECS 84-08524 at Cornell University.
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Avram, F., Taqqu, M.S. Symmetric polynomials of random variables attracted to an infinitely divisible law. Probab. Th. Rel. Fields 71, 491–500 (1986). https://doi.org/10.1007/BF00699038
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DOI: https://doi.org/10.1007/BF00699038
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Symmetric Polynomial
- Triangular Array