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


*

*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? 

*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? 

 A: Something that you perhaps have a better handle on is the time integral of white noise, Brownisn motion. Recall the sample path of Brownian motion is almost surely a continuous, nowhere differentiable function. 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.
