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November, 1980 The Empirical Distribution of Fourier Coefficients
David Freedman, David lane
Ann. Statist. 8(6): 1244-1251 (November, 1980). DOI: 10.1214/aos/1176345197

Abstract

Suppose $X_1, X_2, \cdots$ are independent, identically distributed complex-valued $L^2$ random variables with $EX_1 = 0$ and $E(|X_1|^2) = 1$. Let $Y_{nk}$ be the $k$th Fourier coefficient of $X_1, \cdots, X_n$: $Y_{nk} = \sum^n_{j=1} X_j \exp \big(\frac{2\pi(-1)^{1/2}kj}{n}\big).$ Let $\mu_n$ be the empirical distribution of $\{n^{-1/2}Y_{nk}: k = 1, \cdots, n\}$. Then $\mu_n$ converges to the distribution of $U + iV$, where $U$ and $V$ are independent normal variables with mean 0 and variance $\frac{1}{2}$. This theorem is derived from a similar result for the Fourier coefficients of random permutations of the coordinates of $z^n$, where $z^n$ is a vector with $n$ coordinates such that $\max_k|z^n_k| = o(n^{1/2})$, as $n \rightarrow \infty$.

Citation

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David Freedman. David lane. "The Empirical Distribution of Fourier Coefficients." Ann. Statist. 8 (6) 1244 - 1251, November, 1980. https://doi.org/10.1214/aos/1176345197

Information

Published: November, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0449.62036
MathSciNet: MR594641
Digital Object Identifier: 10.1214/aos/1176345197

Subjects:
Primary: 62E20
Secondary: 42A16

Keywords: complex normal distribution , discrete Fourier transform , Empirical distribution , Fourier coefficients , permutation distribution , Random measures , rankit plot

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 6 • November, 1980
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