  Let L be the class of limiting laws associated with sequences in the second
Wiener chaos. We exhibit a large subset L_0 of L satisfying that, for any
F_infinity in L_0, the convergence of only a finite number of cumulants
suffices to imply the convergence in law of any sequence in the second Wiener
chaos to F_infinity. This result is in the spirit of the seminal paper by
Nualart and Peccati, in which the authors discovered the surprising fact that
convergence in law for sequences of multiple Wiener-It\^o integrals to the
Gaussian is equivalent to convergence of just the fourth cumulant. Also, we
offer analogues of this result in the case of free Brownian motion and double
Wigner integrals, in the context of free probability.
