Suppose given two $n$-degree polynomials $p_1$ and $p_2$. Each coefficient of $p_1$ and $p_2$ is independently sampled from Gaussian distribution with mean 0 and standard deviation $\sigma$, i.e. we sample $n$ times from Gaussian distribution and these sampled values are coefficients of $p_1$. Same goes to $p_2$.

I want to know the distribution of $p$=$p_1\cdot$$p_2$. Does $p$ comply with Gaussian distribution? If so, what is the standard deviation? Are there proofs for this question?

Thank you very much.

  • $\begingroup$ Related : math.stackexchange.com/questions/101062/… $\endgroup$ – Tlön Uqbar Orbis Tertius Feb 2 '17 at 14:29
  • $\begingroup$ By giving this reference, @Tlön Uqbar Orbis Tertius implicitely means that already with constant polynomials, it isn't true. $\endgroup$ – Jean Marie Feb 2 '17 at 14:33
  • $\begingroup$ Otherwise, the seminal works done on the subject of random polynomials in the 1950's are by Mark KAC (see for example (mathworld.wolfram.com/KacFormula.html).) $\endgroup$ – Jean Marie Feb 2 '17 at 14:36
  • $\begingroup$ By the way, reasoning on the distribution of zeros, would maybe another angle of attack of the issue... $\endgroup$ – Jean Marie Feb 2 '17 at 14:39
  • $\begingroup$ "darga"="degree" in which language ? $\endgroup$ – Jean Marie Feb 2 '17 at 14:41

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