I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone be able to give an exact reference? Thank you very much!

"Let $M$ be a $d$ x $d$ primitive integer matrix and $A_i$ a sequence of $d$ x $d$ non-negative integer matrices. Then the image of the positive cone $R^d_{>0}$ under the sequence $\{M A_1 M A_2 \dots M A_k \}_{k=1}^{k=\infty}$ converges weakly to a line."

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone be able to give an exact reference? Thank you very much!

"Let $M$ be a $d$ x $d$ invertible primitive integer matrix and $A_i$ a sequence of $d$ x $d$ invertible non-negative integer matrices. Then the image of the positive cone $R^d_{>0}$ under the sequence $\{M A_1 M A_2 \dots M A_k \}_{k=1}^{k=\infty}$ converges weakly to a line."