  We develop the theoretical basis for the connection of the variety of
anisotropic distributions with the statistical correlations among particles
velocity components. By examining the most common anisotropic distribution, we
derive the correlation coefficient among particle energies, show how this
correlation is connected to the effective dimensionality of the velocity
distribution, and derive the connection between anisotropy and adiabatic
polytropic index. Having established the importance of correlation among
particles in the formulation of anisotropic kappa distributions, we generalize
these distributions within the framework of nonextensive statistical mechanics
and based on the types of homogeneous or heterogeneous correlations among the
particles velocity components. The formulation of the developed generalized
distributions mediates the main two types of anisotropic kappa distributions,
considering (a) equal correlations, or (b) zero correlations, among different
velocity components. Finally, the developed anisotropic kappa distributions are
expressed in terms of the energy and pitch angle in arbitrary reference frames.
