  In this work, we consider a sensor selection drawn at random by a sampling
with replacement policy for a linear time-invariant dynamical system subject to
process and measurement noise. We employ the Kalman filter to estimate the
state of the system. However, the statistical properties of the filter are not
deterministic due to the stochastic selection of sensors. As a consequence, we
derive concentration inequalities to bound the estimation error covariance of
the Kalman filter in the semi-definite sense. Concentration inequalities
provide a framework for deriving semi-definite bounds that hold in a
probabilistic sense. Our main contributions are three-fold. First, we develop
algorithmic tools to aid in the implementation of a matrix concentration
inequality. Second, we derive concentration-based bounds for three types of
stochastic selections. Third, we propose a polynomial-time procedure for
finding a sampling distribution that indirectly minimizes the maximum
eigenvalue of the estimation error covariance. Our proposed sampling policy is
also shown to empirically outperform three other sampling policies: uniform,
deterministic greedy, and randomized greedy.
