  This paper deals with a class of Boltzmann equations on the real line,
extensions of the well-known Kac caricature. A distinguishing feature of the
corresponding equations is that therein, the collision gain operators are
defined by N-linear smoothing transformations. These kind of problems have been
studied, from an essentially analytic viewpoint, in a recent paper by Bobylev,
Cercignani and Gamba [Comm. Math. Phys. 291 (2009) 599-644]. Instead, the
present work rests exclusively on probabilistic methods, based on techniques
pertaining to the classical central limit problem and to the so-called
fixed-point equations for probability distributions. An advantage of resorting
to methods from the probability theory is that the same results - relative to
self-similar solutions - as those obtained by Bobylev, Cercignani and Gamba,
are here deduced under weaker conditions. In particular, it is shown how
convergence to a self-similar solution depends on the belonging of the initial
datum to the domain of attraction of a specific stable distribution. Moreover,
some results on the speed of convergence are given in terms of
Kantorovich-Wasserstein and Zolotarev distances between probability measures.
