  Let $\{\mu(\cdot,t):t\geq0\}$ be the family of probability measures
corresponding to the solution of the inelastic Kac model introduced in
Pulvirenti and Toscani [\textit{J. Stat. Phys.} \textbf{114} (2004) 1453-1480].
It has been proved by Gabetta and Regazzini [\textit{J. Statist. Phys.}
\textbf{147} (2012) 1007-1019] that the solution converges weakly to
equilibrium if and only if a suitable symmetrized form of the initial data
belongs to the standard domain of attraction of a specific stable law. In the
present paper it is shown that, for initial data which are heavier-tailed than
the aforementioned ones, the limiting distribution is improper in the sense
that it has probability 1/2 "adherent" to $-\infty$ and probability 1/2
"adherent" to $+\infty$. It is explained in which sense this phenomenon is
amenable to a sort of explosion, and the main result consists in an explicit
expression of the rate of such an explosion. The presentation of these
statements is preceded by a discussion about the necessity of the assumption
under which their validity is proved. This gives the chance to make an
adjustment to a portion of a proof contained in the above-mentioned paper by
Gabetta and Regazzini.
