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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weak convergence of the empirical characteristic function
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by J. E. Yukich PDF
Proc. Amer. Math. Soc. 95 (1985), 470-473 Request permission

Abstract:

Let $P$ be a probability measure on ${\mathbf {R}}$ such that the density $f(x)$ for $P$ exists and there exists ${x_0} > 0$ such that $f(x) + f( - x)$ is decreasing for all $\left | x \right | \geqslant {x_0}$. Let $c(t)$ be the characteristic function for $P$, ${c_n}(t)$ the empirical characteristic function, and let ${C_n}(t): = {n^{1/2}}({c_n}(t) - c(t))$. New necessary and sufficient metric entropy conditions are obtained for the weak convergence of ${C_n}(t)$ on the space of continuous complex valued functions on $[ - \tfrac {1}{2},\tfrac {1}{2}]$. The result is used to characterize the weak convergence of ${C_n}(t)$ in terms of the tail behavior of $P$ and it also provides the first step towards a generalization of the Borisov-Dudley-Durst theorem. It also provides a partial response to a challenge raised by Dudley.
References
  • I. S. Borisov, On the question of the accuracy of approximation in the central limit theorem for empirical measures, Sibirsk. Mat. Zh. 24 (1983), no. 6, 14–25 (Russian). MR 731039
  • Sándor Csörgő, Limit behaviour of the empirical characteristic function, Ann. Probab. 9 (1981), no. 1, 130–144. MR 606802
  • R. M. Dudley, A course on empirical processes, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 1–142. MR 876079, DOI 10.1007/BFb0099432
  • Mark Durst and Richard M. Dudley, Empirical processes, Vapnik-Chervonenkis classes and Poisson processes, Probab. Math. Statist. 1 (1980), no. 2, 109–115 (1981). MR 626305
  • Michael B. Marcus, Weak convergence of the empirical characteristic function, Ann. Probab. 9 (1981), no. 2, 194–201. MR 606982
  • J. E. Yukich, Laws of large numbers for classes of functions, J. Multivariate Anal. 17 (1985), no. 3, 245–260. MR 813235, DOI 10.1016/0047-259X(85)90083-1
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 95 (1985), 470-473
  • MSC: Primary 60B10; Secondary 60F05
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0806089-1
  • MathSciNet review: 806089