  In this paper, we study a sampling problem where a source takes samples from
a Wiener process and transmits them through a wireless channel to a remote
estimator. Due to channel fading, interference, and potential collisions, the
packet transmissions are unreliable and could take random time durations. Our
objective is to devise an optimal causal sampling policy that minimizes the
long-term average mean square estimation error. This optimal sampling problem
is a recursive optimal stopping problem, which is generally quite difficult to
solve. However, we prove that the optimal sampling strategy is, in fact, a
simple threshold policy where a new sample is taken whenever the instantaneous
estimation error exceeds a threshold. This threshold remains a constant value
that does not vary over time. By exploring the structure properties of the
recursive optimal stopping problem, a low-complexity iterative algorithm is
developed to compute the optimal threshold. This work generalizes previous
research by incorporating both transmission errors and random transmission
times into remote estimation. Numerical simulations are provided to compare our
optimal policy with the zero-wait and age-optimal policies.
