Under reasonable algebraic assumptions and under an infinite second order moment assumption, we show that the logarithm of the norm (log-norm) of a product of random i.i.d. matrices with entries in $\mathbb{R}$ or in any other local field satisfies a generalized Central Limit Theorem (GCLT) in the sense of Paul Lévi. The proof is based on a weak law of large number for the difference $Δ_n$ between the log-norm of the product of the first $n$ matrices and the sum of their log-norms. This weak law of large numbers morally says that $Δ_n$ behaves like a sum of i.i.d. random variables that have a finite moment of order $2q$ as long as the log-norm of each matrices has a finite moment of order $q$ for a given $q > 0$. This gain of moment is the central result of the present paper and is based on the construction of pivotal times. Moreover, these results admit a nice higher rank extension when one looks at the full Cartan projection instead of the log-norm.