# Can we interpret the Fourier transform of binomial coefficients as a random walk ?

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.

• 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).
– user171326
Jul 24, 2017 at 19:24
• Here is the proof.
– user171326
Jul 24, 2017 at 19:29
• @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 ? Jul 24, 2017 at 22:30