  A classical problem of statistical inference is the valid specification of a
model that can account for the statistical dependencies between observations
when the true structure is dense, intractable, or unknown. To address this
problem, a new variance identity is presented, which is closely related to the
Moulton factor. This identity does not require the specification of an entire
covariance structure and instead relies on the choice of two summary constants.
Using this result, a weak law of large numbers is also established for additive
statistics and common variance estimators under very general conditions of
statistical dependence. Furthermore, this paper proves a sharper version of
Hoeffding's inequality for symmetric and bounded random variables under these
same conditions of statistical dependence. Put otherwise, it is shown that,
under relatively mild conditions, finite sample inference is possible in common
settings such as linear regression, and even when every outcome variable is
statistically dependent with all others. All results are extended to estimating
equations. Simulation experiments and an application to climate data are also
provided.
