If I have a random vector $\mathbf{Y}=(Y_1,\dots,Y_n)$ with covariance matrix $\mathbf{\Sigma}$ such that $\mathbf{\Sigma}_{ij}=\gamma(|i-j|)$ (i.e. $\Sigma$ is a Toeplitz matrix), how can I compute $\operatorname{Var}(Y_k\mid Y_1, \dots, Y_{k-1})$? Is there a closed form solution when the distribution of $\mathbf{Y}$ is not known?


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