  In this paper, we consider the problem of distributed parameter estimation in
sensor networks. Each sensor makes successive observations of an unknown
$d$-dimensional parameter, which might be subject to Gaussian random noises.
The sensors aim to infer the true value of the unknown parameter by cooperating
with each other. To this end, we first generalize the so-called dynamic
regressor extension and mixing (DREM) algorithm to stochastic systems, with
which the problem of estimating a $d$-dimensional vector parameter is
transformed to that of $d$ scalar ones: one for each of the unknown parameters.
For each of the scalar problem, both combine-then-adapt (CTA) and
adapt-then-combine (ATC) diffusion-based estimation algorithms are given, where
each sensor performs a combination step to fuse the local estimates in its
in-neighborhood, alongside an adaptation step to process its streaming
observations. Under weak conditions on network topology and excitation of
regressors, we show that the proposed estimators guarantee that each sensor
infers the true parameter, even if any individual of them cannot by itself.
Specifically, it is required that the union of topologies over an interval with
fixed length is strongly connected. Moreover, the sensors must collectively
satisfy a cooperative persistent excitation (PE) condition, which relaxes the
traditional PE condition. Numerical examples are finally provided to illustrate
the established results.
