I wish to know if there is a known relationship between the infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I have found a paper that used the following inequality for the estimation error covariance matrix in Kalman filter. This relationship can be found on Page 4 below Equation 14. The relationship is:
$$E(\|\mathbf{\hat{s}}(k)-\mathbf{s}(k)\|_\infty)\leq\sqrt{\operatorname{tr}(\Sigma(k))},$$
where $\mathbf{\hat{s}}(k)$ is the estimated state vector at time step $k$, $\mathbf{s}(k)$ is the actual state vector, and $\Sigma(k)$ is the estimation error covariance matrix. The Kalman filter equations are taken from the original paper by Kalman.
I could not find any reference for this inequality.
Is this inequality a general relationship between a vector and its covariance matrix?
