Let ${(g{n}){n\\geq 1}}$ be a sequence of independent and identically distributed (i.i.d.) ${d\\times d}$ real random matrices. For ${n\\geq 1}$ set ${G_n = g_n \\ldots g_1}$. Given any starting point ${x=\\mathbb R v\\in\\mathbb{P}^{d-1}}$, consider the Markov chain ${X_n^x \\mathbb G_n v }$ on projective space ${\\mathbb P^{d-1}}$ define norm cocycle by ${\\sigma(G_n, x)= \\log (|G_n v|/|v|)}...