  Wireless Sensor Networks (WSNs) enable a wealth of new applications where
remote estimation is essential. Individual sensors simultaneously sense a
dynamic process and transmit measured information over a shared channel to a
central fusion center. The fusion center computes an estimate of the process
state by means of a Kalman filter. In this paper we assume that the WSN admits
a tree topology with fusion center at the root. At each time step only a subset
of sensors can be selected to transmit observations to the fusion center due to
a limited energy budget. We propose a stochastic sensor selection algorithm
that randomly selects a subset of sensors according to certain probability
distribution, which is opportunely designed to minimize the asymptotic expected
estimation error covariance matrix. We show that the optimal stochastic sensor
selection problem can be relaxed into a convex optimization problem and thus
solved efficiently. We also provide a possible implementation of our algorithm
which does not introduce any communication overhead. The paper ends with some
numerical examples that show the effectiveness of the proposed approach.
