Central limits of squeezing operators

In [1] we have proved a quantum De Moivre-Laplace theorem based on a modification of the Giri-von Waldenfels quantum central limit theorem. In [2] P.A. Meyer outlined a method based on direct calculations which, taking advantage of the explicit structure of the algebra of $2\times 2$ matrices, allows a drastic simplification of the proof of the main result of the first part of our paper and relates it with a similar result obtained, independently and simultaneusly, by Parthasarathy [5]. In the first part of the present note we simplify the Parthasarathy-Meyer method and extend it to deal with arbitrary d-dimensional Bernoulli processes, where $d$ is a natural integer (cfr. Sections (3),(4). We also prove another statement in Meyer's note (cf. Theorem (5.1)). Finally (Section (6)) we show that the method of proof used in [1] allows, with minor modifications, to solve the problem of the central limit approximation of the squeezing states - a problem left open in [1] and to which, due to the nonlinearity of the coupling, Parthasarathy-Meyer direct computational method cannot be applied.