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Is there a article/textbook that treats probability distributions on functions just like basic textbooks cover the classical distributions for scalar variables?

Suppose the random function $f$ has a gaussian process distribution with some mean function $m$ and covariance function $w$. I know how to generate random (functional) deviates from that distribution.

What I'd like to know, for example, is given another function $g$ (that can be evaluated at an arbitrary number of points over the same range than where $f$ is defined), what is the probability that $g$ or a more extreme function was drawn from $f$'s distribution. There must be a way to define a cumulative distribution function and a probability density function for functions!

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Since you can only observe function $g$ at finite number of points $x_1,...,x_n$, then your problem is identical to testing whether the observed vector comes from particular multivariate distribution with given mean and covariance matrix. So if $g(x)=(g(x_1),...,g(x_n))$ comes from multivariate distribution with mean $\mu$ and covariance matrix $\Sigma$ we get

$$(g(x)-\mu)'\Sigma^{-1}(g(x)-\mu)\sim\chi^2_n.$$

Naturally better tests should be available.

Concerning random functions, Kolmogorov theorem states, that the process is fully described by its finite-dimensional distributions. Cumulative distribution function and probability density functions are defined for finite-dimensional distributions, so there you go.

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  • $\begingroup$ can you define finite-distribution please? $\endgroup$
    – yannick
    Dec 13, 2011 at 16:13
  • $\begingroup$ I don't agree with your premise: assume you have a bunch of data points which you just interpolated with a gaussian process. Then the posterior mean and covariance functions fit your data and can be observed at an infinite number of points. $\endgroup$
    – yannick
    Dec 13, 2011 at 16:17
  • $\begingroup$ I meant finite-dimensional distributions. I wrote quickly, sorry for the mistake. As for your second comment, you describe different problem from the one posted in your question. If you have infinite number of observables, then naturally there are more options, but still the end result will be your $g$ plugged in some functional of the process. $\endgroup$
    – mpiktas
    Dec 13, 2011 at 16:55
  • $\begingroup$ I'm sorry I was unclear. I rephrased my question above, I indeed have an infinite number of observables. $\endgroup$
    – yannick
    Dec 14, 2011 at 10:47

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