Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion (monotonicity and convexity). Consider matrices $$ \Sigma_1=[\gamma(i-j)]_{1\le i,j\le m},\quad \Sigma_2=[\gamma(k+i-j)]_{1\le i,j\le m},\quad m>0,~k\ge 0. $$ Show that all entries of $\Sigma_1^{-1}\Sigma_2$ are nonnegative.

Remark: I checked it numerically for different $k$, $m$ and $\alpha$. A weaker statement involving large enough $k\asymp m^{1+\epsilon}$, where $\epsilon>0$ is arbitrarily small, is also of great interest. A further question is to identify general conditions on a positive definite $\gamma(k)$ which fulfills the preceding requirement.

Remark: The problem can be reinterpreted in terms of linear prediction of stationary time series $\{X(n)\}$ whose covariance function is $\gamma(k)$. It is the same as to show that the linear predictor $\hat{X}(k)=\sum_{j=1}^m c_jX(j)$ which minimizes $E |X(k)- \hat{X}(k)|^2$ satisfies $c_j\ge 0$.

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion (monotonicity and convexity). Consider matrices $$ \Sigma_1=[\gamma(i-j)]_{1\le i,j\le m},\quad \Sigma_2=[\gamma(k+i-j)]_{1\le i,j\le m},\quad m>0,~k\ge 0. $$ Show that all entries of $\Sigma_1^{-1}\Sigma_2$ are nonnegative.

Remark: I checked it numerically for different $k$, $m$ and $\alpha$. A weaker statement involving large enough $k\asymp m^{1+\epsilon}$, where $\epsilon>0$ is arbitrarily small, is also of great interest. A further question is to identify general conditions on a positive definite $\gamma(k)$ which fulfills the preceding requirement.

Remark: The problem can be reinterpreted in terms of linear prediction of stationary time series $\{X(n)\}$ whose covariance function is $\gamma(k)$. It is the same as to show that the linear predictor $\hat{X}(k)=\sum_{j=1}^m c_jX(j)$ which minimizes the mean square error $E |X(k)- \hat{X}(k)|^2$ satisfies $c_j\ge 0$.

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion (monotonicity and convexity). Consider matrices $$ \Sigma_1=[\gamma(i-j)]_{1\le i,j\le m},\quad \Sigma_2=[\gamma(k+i-j)]_{1\le i,j\le m},\quad m>0,~k\ge 0. $$ Show that all entries of $\Sigma_1^{-1}\Sigma_2$ are nonnegative.

Remark: I checked it numerically for different $k$, $m$ and $\alpha$. A weaker statement involving large enough $k\asymp m^{1+\epsilon}$, where $\epsilon>0$ is arbitrarily small, is also of great interest. A further question is to identify general conditions on a positive definite $\gamma(k)$ which fulfills the preceding requirement.

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion (monotonicity and convexity). Consider matrices $$ \Sigma_1=[\gamma(i-j)]_{1\le i,j\le m},\quad \Sigma_2=[\gamma(k+i-j)]_{1\le i,j\le m},\quad m>0,~k\ge 0. $$ Show that all entries of $\Sigma_1^{-1}\Sigma_2$ are nonnegative.

Remark: I checked it numerically for different $k$, $m$ and $\alpha$. A weaker statement involving large enough $k\asymp m^{1+\epsilon}$, where $\epsilon>0$ is arbitrarily small, is also of great interest. A further question is to identify general conditions on a positive definite $\gamma(k)$ which fulfills the preceding requirement.

Remark: The problem can be reinterpreted in terms of linear prediction of stationary time series $\{X(n)\}$ whose covariance function is $\gamma(k)$. It is the same as to show that the linear predictor $\hat{X}(k)=\sum_{j=1}^m c_jX(j)$ which minimizes $E |X(k)- \hat{X}(k)|^2$ satisfies $c_j\ge 0$.