Let $X_{t_1}, \ldots , X_{t_n}$ be sampled from a zero-mean random field $(X_t)_{t \in \mathbb{R}^d}$ at $n$ locations $t_i \in \mathbb{R}^d$, $i=1, \ldots , n$ and collect them in the $n$-vector $Z$. Write $Z=\text{Cov}(Z)$ and let $K = (K_i)_{i=1}^n$ be the $n \times 1$ vector with entries $K_i = \rho(t_i, t_0)$.

Consider the case where some eigenvalue of $\Sigma$ is zero. Then I want to show that $K$ is in the column space of $\Sigma$.

A provided hint suggests to take $0 \neq a \in \mathbb{R}^d$ for which $a' \Sigma a = 0$ and consider $\text{Cov}(c'Z,a'Z)$ and $\text{Cov}(X_{t_0},a'Z)$.

However, I do not see how this would prove the above statement. How do this $\text{Cov}(c'Z,a'Z)$ and $\text{Cov}(X_{t_0},a'Z)$ relate to the column space of $\Sigma$? Any help is appreciated!


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