  The Allan Variance (AV) is a widely used quantity in areas focusing on error
measurement as well as in the general analysis of variance for autocorrelated
processes in domains such as engineering and, more specifically, metrology. The
form of this quantity is widely used to detect noise patterns and indications
of stability within signals. However, the properties of this quantity are not
known for commonly occurring processes whose covariance structure is
non-stationary and, in these cases, an erroneous interpretation of the AV could
lead to misleading conclusions. This paper generalizes the theoretical form of
the AV to some non-stationary processes while at the same time being valid also
for weakly stationary processes. Some simulation examples show how this new
form can help to understand the processes for which the AV is able to
distinguish these from the stationary cases and hence allow for a better
interpretation of this quantity in applied cases.
