I have a question regarding a time changed Gaussian process. Suppose that $\{X_t\}_{t\in \mathbb{R}}$ is a Gaussian process with $0$ mean and covariance function $K$.
Suppose that $\phi$ is a deterministic function i.e. $\phi:\mathbb R \to \mathbb R$. What conditions should we have on the function $\phi$ such that $\{X_{\phi(t)}\}_{t\in \mathbb{R}}$ is still a Gaussian process ?
As you said $\phi$ is a deterministic function, there should be no further condtions be necessary.
Because a time change is a nondecreasing cadlag function and Gaussian process is defined as (thats the definition I learned) a stochastic process $X_t$, so that for every $n\in\mathbb N$, every $t_1<...<t_n$ and every linear mapping $$l:\mathbb R^n\to \mathbb R$$ holds, that the real RV $l(X_{t_1},...,X_{t_n})$ is normal distributed. A deterministic change of time $\phi$ does not change anything of this property, since you can just choose new times $t'_1=\phi(t_1),...,t'_n=\phi(t_n)$ of which you already know, that $l(X_{t'_1},...,X_{t'_n})$ ist normal distributed.
If your function is not deterministic, this would not be the case in general as this post shows.
