Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X_t:\Omega \times \mathbb [0,T] \rightarrow \mathbb R$ a continuous stochastic process with the following property:
There exists an $n>1$ such that for any real sequence $0<s_1<\ldots<s_n<t<T$, denoting $\mathbb X=(X_{s_1},\ldots, X_{s_n})$ and $\mathbb A_i= Cov[X_{s_i},\mathbb X]$, we have $Cov[X_{t},\mathbb X]= \sum\limits_{i=1}^n a_i \mathbb A_i$ for some real coefficients $a_i \neq 0 $ for all $i=1,\ldots, n$, and if we remove any term from $\sum\limits_{i=1}^n a_i \mathbb A_i$, then the equation has no solutions anymore.
Does such stochastic process exist ? If so, can you give an example ? If not, why ?
For $n=1$, the Brownian bridge has this property. I would be very happy to find such a process for $n=2$, and over the moon if we can build such a process for any $n$.
EDIT: In the case of Brownian bridge (so $n=1$), we have (writing $s_1$ as $s$)
$Cov[X_{t},\mathbb X]=Cov[X_{t},X_s]$ and $\mathbb A_1= Cov[X_{s}, X_s]$. We get
$$Cov[X_{t},\mathbb X]= \frac{T-t}{T-s} \mathbb A_1$$
so $a_1=\frac{T-t}{T-s}$ which is never $0$.