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So suppose I have some random function distributed as a Gaussian process on say, the interval $[0,1]$ with mean function $\mu$ and covariance function $\Lambda$. When is there a nice formula for the probability that the $\arg \max$ of the random function lies in some sub-interval $[a,b]$? Obviously there are trivial cases, like where the mean function is zero the covariance function is translation invariant, but what about other cases? Any help appreciated, even if it just pointing me to resources that might help, thanks.

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First note that if we define a random variable $M = \text{arg max}_t \Lambda(t)$, then your question is equivalent to asking when does the cumulative distribution function of $M$ have a nice formula. One nontrivial case where this happens is when $\Lambda$ is a Brownian motion (see the third arcsine law).

Although not directly applicable, it may be useful to look into the list of examples given in the introduction to this paper, which describes cases where the distribution of the max (as opposed to argmax) is known. This paper provides a connection between the distribution of the argmax and the distribution of the max of related processes, although I do not know if it will help in computing specific cases or not.

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  • $\begingroup$ The paper seems to give examples where we know the distribution of the maximum, not the argmax. The examples where we have formulas for the argmax may be different. $\endgroup$ – user54038 Dec 16 '18 at 18:40
  • $\begingroup$ @user54038, You are quite right. I have now edited my answer to make this clear. $\endgroup$ – Brent Kerby Dec 17 '18 at 15:48

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