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We know that:

  1. binomial coefficients $\tbinom {n}{k}$ will approach normal distribution in the limit.
  2. binomial coefficients can be interpreted as a one-dimensional random walk.
  3. Fourier transform of a standard normal distribution is a standard normal distribution.

My questions are:

Q1: Before we take the limit for binomial coefficients, that is, for finite (discrete) binomial coefficients $\tbinom {n}{k}$, if we take discrete Fourier transform(DFT) of it, can we interpret the result of each step for such DFT as another one-dimensional random walk ?

In a broader sense, I am trying to seek "deep" connection between one-dimensional random walk with Fourier transform. What are such "deep" connection ?

Thank you.

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  • $\begingroup$ There is a proof that $\Bbb P\{\exists k > 0, X_k = 0\} = 1$ using the Fourier transform (I don't remember the details but I can search if you are interested). $\endgroup$
    – user171326
    Jul 24, 2017 at 19:24
  • $\begingroup$ Here is the proof. $\endgroup$
    – user171326
    Jul 24, 2017 at 19:29
  • $\begingroup$ @N.H. Thank you for the reply. My main question is, for finite (discrete) binomial coefficients $\tbinom {n}{k}$, if we take discrete Fourier transform(DFT) of it, can we interpret the result of each step for such DFT as another one-dimensional random walk ? $\endgroup$
    – david
    Jul 24, 2017 at 22:30

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