Let g_1,\ldots,g_n,\ldots be random iid invertible matrices of size d\geq 2. There is a general philosophy (due to Furstenberg, Kesten, Guivarc'h, Le Page, Raugi,...) that the process \log(|g_n\cdots g_1|) follows the same limit laws as a classical random walk on \mathbb R (where |g| denotes the operator norm of a matrix g). In this talk, I will respect this philosophy by discussing limit laws for the process \log(|g_n\cdots g_1|)conditioned to stay non-negative. This is a joint work with I. Grama and H. Xiao.