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How can one simulate a Gaussian process with nonzero covariance?

I've never simulated a process. I have searched for some basic examples, but I cannot make sense of all of it. Can someone give an an algorithm (continuous and discrete) followed with a minor explanation of such a simulation? How can the Fourier transform be used?

Iced Palmer
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  • Closely related to that particular covariance function: http://stats.stackexchange.com/questions/29091, though this, too, is for a set of points. Can you clarify what you expect a "simulation over the real line" to output - it's (obviously?) not possible to actually either store or generate an infinite number of values (one $f(t)$ for every $t$) with a computer, but perhaps you are after something else – Juho Kokkala Oct 19 '16 at 05:43
  • @ Juho Kokkala I expect to use the fourier transform. For the real line, I am looking or a more theoretical way to proceed rather than using a computer. Is there a theoretical way to simulate over the line? – Iced Palmer Oct 19 '16 at 05:48
  • @ Juho Kokkala also, regarding that example you cited. That looks like a discrete version. Can you give a continuous version? – Iced Palmer Oct 19 '16 at 05:50
  • I don't know what a "theoretical way to simulate" would be - or, what tools are available - apparently the definition of the process itself is not allowed as an oracle for sampling from the process, but what is? Besides sampling the process in a finite set of points, sampling from a truncated basis function decomposition comes into mind but I'm not sure if that, either, is what you are after (my intuition would anyway be that a stationary process cannot be very accurately sampled over the entire real line with any method based on a finite number of random numbers) – Juho Kokkala Oct 19 '16 at 06:06
  • I've edited the question, can the hold be removed? – Iced Palmer Oct 20 '16 at 06:07
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    since a Brownian motion is an integral of a gaussian process you can (in `R`) simply use `diff(cumsum(rnorm(1000)))` for each varaible. I.e. the difference between "time steps" of a cummulative sum of a gaussian variable. To use a multivariate gaussian sampling use `rmvnorm` from `mvtnorm` package. – Drey Oct 20 '16 at 12:54

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