## Annals of Statistics

- Ann. Statist.
- Volume 8, Number 6 (1980), 1244-1251.

### The Empirical Distribution of Fourier Coefficients

#### 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 $n \rightarrow \infty$.

#### Article information

**Source**

Ann. Statist., Volume 8, Number 6 (1980), 1244-1251.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345197

**Digital Object Identifier**

doi:10.1214/aos/1176345197

**Mathematical Reviews number (MathSciNet)**

MR594641

**Zentralblatt MATH identifier**

0449.62036

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series {For automorphic theory, see mainly 11F30}

**Keywords**

Fourier coefficients empirical distribution discrete Fourier transform random measures complex normal distribution permutation distribution rankit plot

#### Citation

Freedman, David; lane, David. The Empirical Distribution of Fourier Coefficients. Ann. Statist. 8 (1980), no. 6, 1244--1251. doi:10.1214/aos/1176345197. https://projecteuclid.org/euclid.aos/1176345197