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Weak convergence of the empirical characteristic function


Author: J. E. Yukich
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
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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\vert x \right\vert \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 [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0806089-1
Keywords: Empirical characteristic function, central limit theorem on $ C([ - \tfrac{1}{2},\tfrac{1}{2}])$
Article copyright: © Copyright 1985 American Mathematical Society

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