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MLE of the covariance function for a Gaussian process
My professor posted a question on which I had no answer. Maybe you can help me.
Let $S$ be a random element of the space of all right-continuous functions on $[0, 1]$ (using the inner product defined by integration over $[0, 1]$). We can assume that $S$ is Gaussian with covariance function $c$ (i.e., the integral defining the covariance is $0$). The random function is $f \colon [0, 1] \to \mathbb{R}$. What is the MLE for $c$?
A:
I believe I figured it out.
The necessary assumptions are that the random function is Gaussian (I am not 100% certain about that), and that $f$ has the property that $f(0) = 0$. By the continuity of $f$, the constant is removed from the variance of the random function. So we can assume that $f \col
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