In a hidden Markov model (HMM) we have a process $X_k$ that evolves according to:
$$ X_{k+1} = X_k + W_{k+1}, \quad W_{k+1} \sim N(0, \sigma_{W}^2), $$
where $\{W_k \}$ are IID and $X_0 = W_0$. We can only observe noisy measurements $Y_k$ of the process $X_k$,
$$ Y_k = X_k + V_k, \quad V_k \sim N(0, \sigma_{V}^2).$$
It's easy to see that $f(k) = X_k = \sum_{i=0}^{k+1}W_i$ is a Gaussian process with mean $m(n) = 0$ and covariance kernel $k(n,m) = (1+\min(n,m))\sigma_{W}^2.$
Now consider the the posterior Gaussian process $f(n) = X_n | Y_1= y_1, \dots Y_k = y_k.$ What is the mean and covariance kernel for this process?
