On this wikipedia page it is written that

Brown noise can be produced by integrating white noise

What kind of white noise is meant? The Gaussian or uniform white noise?


1 Answer 1


White noise is noise that has equal (uniform) amplitude across all frequencies. When we say "white" we're talking about the power spectral density (PSD) of the noise.

Saying something like "Gaussian noise" means the statistical properties of any one sample of the noise is distributed Gaussian. You can actually have Cauchy, Poisson, Gaussian etc. distributions that define any one sample of the noise. For any distribution, however, if the power spectral density of the noise is uniform, that noise is white.

So in the integral defining the Weiner process: $$W(t) = \int_0^t\frac{dW(\tau)}{d\tau}{d\tau}$$

$W(t)$ has samples with Cauchy, Poisson, Gaussian, etc. distributions as long as the power spectral density is uniform, making it white noise.

  • 1
    $\begingroup$ I think the OP wanted to ask (or I think the answer should be) is that how different distributions of $W(t)$ does not change the property defining Brownian noise, i.e. $\textrm{PSD} \propto 1/f^2$. $\endgroup$
    – AlexTP
    May 11, 2017 at 16:27
  • $\begingroup$ @AlexTP That's definitely a good point! I did sort of leave OP to his/her own devices to infer from the result in the wiki. Hopefully these comments will fill the gap. Thanks Alex. $\endgroup$
    – Envidia
    May 11, 2017 at 16:31

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