Suppose we know that $X$ is a stationary mean zero gaussian process with known parameters. Suppose an experiment provides me with a collection of samples $(t_i, x_i)$ for $i = 1, 2, . . . , N$. How could I compute a p-value for the hypothesis that the $x_i$ are the values of $X$ at the times $t_i$?.
I was thinking of using the cholesky decomposition of the covariance matrix $\Sigma=LL^T$ to get iid random variables, i.e., $Y=L^TX \sim N(0,I)$ but not too confident on how to proceed. How can I compute the p-value for the specified test?
If knowing the parameters of $X$ means knowing the mean function $\mu(t)$ and the covariance kernel $k(t,t')$ then we can easily test for the values $(t_i,x_i)$. In fact, by consistency, we know that $X(t_i) = N(\mu(t_i), k(t_i,t_i))$ so we only have to test if $x_i$ comes form $N(\mu(t_i), k(t_i,t_i))$ for each $i = 1,...,N$.