  Remote estimation with an energy harvesting sensor with a limited data and
energy buffer is considered. The sensor node observes an unknown Gaussian field
and communicates its observations to a remote fusion center using the energy it
harvested. The fusion center employs minimum mean-square error (MMSE)
estimation to reconstruct the unknown field. The distortion minimization
problem under the online scheme, where the sensor has access to only the
statistical information for the future energy packets is considered. We provide
performance bounds on the achievable distortion under a slotted block
transmission scheme, where at each transmission time slot, the data and the
energy buffer are completely emptied. Our bounds provide insights to the
trade-offs between the buffer sizes, the statistical properties of the energy
harvesting process and the achievable distortion. In particular, these
trade-offs illustrate the insensitivity of the performance to the buffer sizes
for signals with low degree of freedom and suggest performance improvements
with increasing buffer size for signals with relatively higher degree of
freedom. Depending only on the mean, variance and finite support of the energy
arrival process, these results provide practical insights for the battery and
buffer sizes for deployment in future energy harvesting wireless sensing
systems.
