  We study stochastic properties of the norm cocycle associated with iid
products of positive matrices. We obtain the almost sure invariance principle
(ASIP) with rate o(n 1/p) under the optimal condition of a moment or order p >
2 and the Berry-Esseen theorem with rate O(1/ $\sqrt$ n) under the optimal
condition of a moment of order 3. The results are also valid for the matrix
norm. For the matrix coefficients, we also have the ASIP but we obtain only
partial results for the Berry-Esseen theorem. The proofs make use of coupling
coefficients that surprisingly decay exponentially fast to 0 while there is
only a polynomial decay in the case of invertible matrices. All the results are
actually valid in the context of iid products of matrices leaving invariant a
suitable cone.
