  We consider distributed recursive estimation of consensus+innovations type in
the presence of heavy-tailed sensing and communication noises. We allow that
the sensing and communication noises are mutually correlated while independent
identically distributed (i.i.d.) in time, and that they may both have infinite
moments of order higher than one (hence having infinite variances). Such
heavy-tailed, infinite-variance noises are highly relevant in practice and are
shown to occur, e.g., in dense internet of things (IoT) deployments. We develop
a consensus+innovations distributed estimator that employs a general
nonlinearity in both consensus and innovations steps to combat the noise. We
establish the estimator's almost sure convergence, asymptotic normality, and
mean squared error (MSE) convergence. Moreover, we establish and explicitly
quantify for the estimator a sublinear MSE convergence rate. We then quantify
through analytical examples the effects of the nonlinearity choices and the
noises correlation on the system performance. Finally, numerical examples
corroborate our findings and verify that the proposed method works in the
simultaneous heavy-tail communication-sensing noise setting, while existing
methods fail under the same noise conditions.
