  Neutrinoless double-beta decay nuclear transition matrix elements are
generated by an effective two-body transition operator and it consists of
Gamow-Teller like and Fermi like (also tensor) operators. Spectral distribution
method for the corresponding transition strengths (squares of the transition
matrix elements) involves convolution of the transition strength density
generated by the non-interacting particle part of the Hamiltonian with a
spreading function generated by the two-body part of the Hamiltonian. Extending
the binary correlation theory for spinless embedded $k$-body ensembles to
ensembles with proton-neutron degrees of freedom, we establish that the
spreading function is a bivariate Gaussian for transition operators
$\co(k_\co)$ that change $k_\co$ number of neutrons to $k_\co$ number of
protons. Towards this end, we have derived the formulas for the fourth-order
cumulants of the spreading function and calculated their values for some heavy
nuclei; they are found to vary from $\sim -0.4$ to -0.1. Also for nuclei from
$^{76}$Ge to $^{238}$U, the bivariate correlation coefficient is found to vary
from $\sim 0.6 - 0.8$ and these values can be used as a starting point for
calculating nuclear transition matrix elements using the spectral distribution
method.
