It is well known that the distribution of sums of random variables converges to Gaussian. In this post, we look look at the convergence of Fourier transform of uniformly distributed random variables.
Exact and analytical formulas for convergence are interesting as they help develop an intuition about the asymptotic convergence phenomenon and understand the convergence error for finite number of samples. Under the scanner today is the convergence behaviour of Fourier transform of uniformly distributed random variables (URVs).
Let denote complex-valued zero mean i.i.d URVs with finite support for . Their Fourier transform is given by
(2.36) |
where are the roots of unity, and .
Let , and . Variable preserves the distribution of . Therefore, we have the characteristic function of the real part of the Fourier transformed variable as
(2.37) |
Interestingly, for , the limits of the derivatives , for , exist! Therefore, the Taylor expansion of around is given by
(2.38) |
The Taylor series expansion given above may be easily verified using symbolic solvers like Wolfram Alpha Online.
Substituting the Taylor series expansion in the characteristic function above, we have (only the first two terms of the Taylor series are shown)
(2.39) |
Therefore, as ,
(2.40) |
where
(2.41) |
The right hand side expression in (2.40) showing the asymptotic convergence of the characteristic function of is readily identified as the characteristic function of Gaussian distribution with zero mean and variance . A similar expression may also be found for in a straightforward manner.
Therefore, the exact convergence formula for the Fourier transform of URVs is as shown in (2.39)!
ARK
First published: 25th Jun. 2016 on aravindhk-math.blogspot.com
Modified: 17th Dec. 2023 – Style updates for LaTeX
Modified: 30th Dec. 2023 – Minor updates to text