  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.
They aim to infer 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, an estimation scheme is given, where each sensor fuses the
regressors and measurements in its in-neighborhood and updates its local
estimate by using least-mean squares. Particularly, a counter is also
introduced for each sensor, which prevents any (noisy) measurement from being
repeatedly used such that the estimation performance will not be greatly
affected by certain extreme values. A novel excitation condition termed as
\textit{local persistent excitation} (Local-PE) condition is also proposed,
which relaxes the traditional persistent excitation (PE) condition and only
requires that the collective signals in each sensor's in-neighborhood are
sufficiently excited. With the Local-PE condition and proper step sizes, we
show that the proposed estimator guarantee that each sensor infers the true
parameter in mean square, even if any individual of them cannot. Numerical
examples are finally provided to illustrate the established results.
