  It is known that the joint limit distribution of independent Wigner matrices
satisfies a very special asymptotic independence, called freeness. We study the
joint convergence of a few other patterned matrices, providing a framework to
accommodate other joint laws. In particular, the matricial limits of symmetric
circulants and reverse circulants satisfy, respectively, the classical
independence and the half independence. The matricial limits of Toeplitz and
Hankel matrices do not seem to submit to any easy or explicit
independence/dependence notions. Their limits are not independent, free or half
independent.
