fourier series with random phases

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Let $X$ be uniformly distributed on $[0,1]$ and consider the sequence of functions $f_n(X) = sin(2npi X)$. Let $phi:Omegarightarrow [0,2pi]$ be a uniformly distributed random variable. Let $phi_n=phi(omega_n)$ for $omega_n$ all distinct. Then apparently, for large $N$, $(2/N)^1/2sum_n=1^N f_n(X-phi_n)$ will then be approximately normally distributed with mean zero and variance one for almost all realizations of the phases. Yet as soon as $phi_n$, $nleq N$, are realized, $f_n(X-phi_n)$, $nleq N$, are not independent random variables and yet they still obey the conclusion of the central limit theorem. Why?

I want to make the following conjecture:
The probability distribution function of $s_N;phi_1,dots, phi_N(X) = (2/N)^1/2sum_n=1^N f_n(X-phi_n)$ is asymptotic to the standard normal distribution function as $Nrightarrow infty$ for almost all vectors $(phi_1,dots,phi_N)in[0,2pi]^N$, the sequence being interpreted as $(phi_1), (phi_1, phi_2),...(phi_1,phi_2,...,phi_N),...$. Is this sensible?

If it is true and the proof is trivial, I would appreciates hints on how to show it more so than the full proof.

fourier-series central-limit-theorem

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edited Jul 27 at 14:49

asked Jul 25 at 19:35

itheypsi

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up vote
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Let $X$ be uniformly distributed on $[0,1]$ and consider the sequence of functions $f_n(X) = sin(2npi X)$. Let $phi:Omegarightarrow [0,2pi]$ be a uniformly distributed random variable. Let $phi_n=phi(omega_n)$ for $omega_n$ all distinct. Then apparently, for large $N$, $(2/N)^1/2sum_n=1^N f_n(X-phi_n)$ will then be approximately normally distributed with mean zero and variance one for almost all realizations of the phases. Yet as soon as $phi_n$, $nleq N$, are realized, $f_n(X-phi_n)$, $nleq N$, are not independent random variables and yet they still obey the conclusion of the central limit theorem. Why?

I want to make the following conjecture:
The probability distribution function of $s_N;phi_1,dots, phi_N(X) = (2/N)^1/2sum_n=1^N f_n(X-phi_n)$ is asymptotic to the standard normal distribution function as $Nrightarrow infty$ for almost all vectors $(phi_1,dots,phi_N)in[0,2pi]^N$, the sequence being interpreted as $(phi_1), (phi_1, phi_2),...(phi_1,phi_2,...,phi_N),...$. Is this sensible?

If it is true and the proof is trivial, I would appreciates hints on how to show it more so than the full proof.

fourier-series central-limit-theorem

share|cite|improve this question

edited Jul 27 at 14:49

asked Jul 25 at 19:35

itheypsi

385

add a comment | 

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

Let $X$ be uniformly distributed on $[0,1]$ and consider the sequence of functions $f_n(X) = sin(2npi X)$. Let $phi:Omegarightarrow [0,2pi]$ be a uniformly distributed random variable. Let $phi_n=phi(omega_n)$ for $omega_n$ all distinct. Then apparently, for large $N$, $(2/N)^1/2sum_n=1^N f_n(X-phi_n)$ will then be approximately normally distributed with mean zero and variance one for almost all realizations of the phases. Yet as soon as $phi_n$, $nleq N$, are realized, $f_n(X-phi_n)$, $nleq N$, are not independent random variables and yet they still obey the conclusion of the central limit theorem. Why?

I want to make the following conjecture:
The probability distribution function of $s_N;phi_1,dots, phi_N(X) = (2/N)^1/2sum_n=1^N f_n(X-phi_n)$ is asymptotic to the standard normal distribution function as $Nrightarrow infty$ for almost all vectors $(phi_1,dots,phi_N)in[0,2pi]^N$, the sequence being interpreted as $(phi_1), (phi_1, phi_2),...(phi_1,phi_2,...,phi_N),...$. Is this sensible?

If it is true and the proof is trivial, I would appreciates hints on how to show it more so than the full proof.

fourier-series central-limit-theorem

share|cite|improve this question

edited Jul 27 at 14:49

asked Jul 25 at 19:35

itheypsi

385

Let $X$ be uniformly distributed on $[0,1]$ and consider the sequence of functions $f_n(X) = sin(2npi X)$. Let $phi:Omegarightarrow [0,2pi]$ be a uniformly distributed random variable. Let $phi_n=phi(omega_n)$ for $omega_n$ all distinct. Then apparently, for large $N$, $(2/N)^1/2sum_n=1^N f_n(X-phi_n)$ will then be approximately normally distributed with mean zero and variance one for almost all realizations of the phases. Yet as soon as $phi_n$, $nleq N$, are realized, $f_n(X-phi_n)$, $nleq N$, are not independent random variables and yet they still obey the conclusion of the central limit theorem. Why?

I want to make the following conjecture:
The probability distribution function of $s_N;phi_1,dots, phi_N(X) = (2/N)^1/2sum_n=1^N f_n(X-phi_n)$ is asymptotic to the standard normal distribution function as $Nrightarrow infty$ for almost all vectors $(phi_1,dots,phi_N)in[0,2pi]^N$, the sequence being interpreted as $(phi_1), (phi_1, phi_2),...(phi_1,phi_2,...,phi_N),...$. Is this sensible?

If it is true and the proof is trivial, I would appreciates hints on how to show it more so than the full proof.

fourier-series central-limit-theorem

share|cite|improve this question

edited Jul 27 at 14:49

asked Jul 25 at 19:35

itheypsi

385

share|cite|improve this question

share|cite|improve this question

edited Jul 27 at 14:49

edited Jul 27 at 14:49

edited Jul 27 at 14:49

asked Jul 25 at 19:35

itheypsi

385

asked Jul 25 at 19:35

itheypsi

385

asked Jul 25 at 19:35

itheypsi

385

385

add a comment | 

add a comment | 

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