  Second order circularity, also called properness, for complex random
variables is a well known and studied concept. In the case of quaternion random
variables, some extensions have been proposed, leading to applications in
quaternion signal processing (detection, filtering, estimation). Just like in
the complex case, circularity for a quaternion-valued random variable is
related to the symmetries of its probability density function. As a
consequence, properness of quaternion random variables should be defined with
respect to the most general isometries in $4D$, i.e. rotations from $SO(4)$.
Based on this idea, we propose a new definition of properness, namely the
$(\mu_1,\mu_2)$-properness, for quaternion random variables using invariance
property under the action of the rotation group $SO(4)$. This new definition
generalizes previously introduced properness concepts for quaternion random
variables. A second order study is conducted and symmetry properties of the
covariance matrix of $(\mu_1,\mu_2)$-proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.
