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I have two questions about a Gaussian white noise $\xi_t$:

  1. First of all, $\xi_t$ is a white noise. So, it's variance should be infinitive. But if the process is Gaussian then it has finite variance. How can Gaussian white noise exist?
  2. Say, we have to compute the function $u(t)= \sigma \xi_t$ at the moment $t=t_i$ during an integration of another complex system. If $\xi_t$ is a standard Gaussian white noise then we could simulate $\xi_t$ as a random number of standard normal distribution. And we get $u(t_i)=\sigma \hat{\xi_t},\, \hat{\xi_t} \in N(0,1)$. But in case the process isn't a Gaussian one we could write as follows: $$ u(t) dt =\sigma dW,$$ where W - a Weiner process. Then we get $u(t_i)=\frac{\sigma}{\sqrt{\Delta t}}\hat{\xi_t}, \, \hat{\xi{_t}} \in N(0,1).$
    Where do I make mistake?
Artem Zefirov
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  • I'm not sure what you mean by your second question. How is your example not a Gaussian process? And what contradiction are you trying to point out? In general, switching freely between continuous in real time is probably what's screwing you up here since thr heart of this is all about subtleties of the continuum limit. – spaceisdarkgreen Sep 27 '17 at 18:36
  • In the second question I want to realize how to simulate a white noise which is not Gaussian. As a variance of a white noise is infinite, the second approach looks like quite natural: if $\Delta t =t_{i+1}-t_i \longrightarrow 0$ then at the moment $t=t_i$ the variable $u(t_i)$ goes to infinite. But I'm still confused about it. Both question, probably, are about the same problem. – Artem Zefirov Sep 27 '17 at 18:56
  • well that last part comports well with your first question where $"\xi(t)"$ has infinite variance. If you read my answer you will see why there is no such thing as $\xi(t)$. – spaceisdarkgreen Sep 27 '17 at 19:08

1 Answers1

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Something that you perhaps have a better handle on is the time integral of white noise, Brownian motion. Recall the sample paths of Brownian motion are almost surely continuous and nowhere differentiable. Since a "sample path" of white noise is the derivative of a sample path of Brownian motion, it is the derivative of a nowhere differentiable function. This doesn't bode well.

So the "sample path" of white noise is something whose integral makes sense but doesn't really make sense as a function itself. In other words, it's a distribution / generalized function, like the Dirac delta.

So when you write down discrete time Gaussian white noise (which is just independent Gaussians with zero mean and some finite variance - perfectly sensible) and then carefully try to make sense of it in continuous time you find that the covariance function is a Dirac delta. If you interpret the sample path as a bunch of random variables, one for each time, you find nonsensically that their variance is infinite. This tells you that it doesn't make sense to think of the sample path as a function, which is the same thing we said before. It's just math telling you the same thing in two different ways.

Lance
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spaceisdarkgreen
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  • So, as I've understand, a "white noise" itself tells nothing about random variables distribution at each discrete time. That's why we need "Gaussian" to handle on these random variables. In conclusion, the first approach of a Gaussian white noise simulation from the second question is correct, and the second one is wrong because I'm trying to go to infinitely small time thought the derivative in a common sense of a Weiner process doesn't exist. And if the process is just white noise then we can not simulate it at a discrete time. Right? – Artem Zefirov Sep 27 '17 at 19:47
  • i think you are getting some of it. In continuous time, in a sense the "Gaussian" does not matter thanks to the central limit theorem... start with any zero mean finite variance rv on a small scale and add them up and you get regular old Gaussian wiener process macroscopically. If you want Gaussian or nongaussian white noise in discrete time just simulate independent rvs with the distribution of your choice. – spaceisdarkgreen Sep 27 '17 at 20:07
  • Your first sentence leaves me uneasy though. What I'm saying is when you're talking about continuous time white noise (which I assumed you were cause of the remark about infinite variance) the "random variables distribution at each discrete time" is not a concept that even makes sense... these random variables do not exist. None of this is a problem in discrete time. You are free to take a sequence of independent rvs from any distribution you like. So it both makes sense to talk about the random variable and to decide on its distribution, Gaussian or otherwise. – spaceisdarkgreen Sep 27 '17 at 20:31
  • I misunderstand you. We can fix time. And at this moment we can deal with a random variable. Why do you say, that it doesn't exist? And I don't quite understand, as I see now, why one can use arbitrary distribution in order to get Gaussian white noise. That is, how do you apply the central limit theorem? We don't summarize anything here. – Artem Zefirov Sep 27 '17 at 21:49
  • I say it does not exist because we cannot do that and it literally does not exist. If it existed, it would be the derivative at $t$ of a nowhere differentiable function... can't happen. I did not say we can use an arbitrary distribution to get gaussian wn, I said an arbitrary dist would add up to a (gaussian) Brownian motion. It's imprecise as I said it, but the idea is that if you take a discrete random walk with any finite variance step distribution, at long timescales it will look like Brownian motion because the central limit says the sum converges in distribution to a normal. – spaceisdarkgreen Sep 27 '17 at 23:57
  • I still can't understand how to deal with a discrete time white noise. If it doesn't exist then direct simulation at each time doesn't make sense at all. Then I should write something like: $u_{i+1}-u_i= \sigma \sqrt{\Delta t} \hat{\xi_t},, \hat\xi_t \in N(0,1)$. Could you formalize a bit your point? – Artem Zefirov Sep 28 '17 at 07:04
  • A discrete time white noise exists and makes perfect sense, as I've said before. It's continuous time white noise that doesn't exist as a function. I don't know what you're asking or trying to do. $u_{i+1}-u_i = \sigma\sqrt{\Delta t}\xi_i$ is an equation for Euler method simulation of brownian motion (whose sample paths do exist as continuous time functions and values at different times can be thought of as random variables). Of course you cannot actually produce a continuous-time process, only a discrete-time approximation to it. Could you clarify what you're asking? – spaceisdarkgreen Sep 28 '17 at 13:03