Ergodic theory reference for converging sequences of matrices

Question

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."

Answer 1

Here is a result of a similar, but not exactly the same, nature. Suppose $M$ is a strictly positive matrix (that is, all entries are real and strictly positive) and none of the nonnegative-entried $A_i$ have a row of zeros or a column of zeros. Then the sequence $(MA_i)$ is weakly ergodic (which is, I think, what you mean by converges to a line?); that is, there is a unique harmonic function on the system. This is a consequence of results in section 5 of

D Handelman [me], Eigenvectors and ratio limit theorems for Markov chains and their relatives, Journal d’Analyse Mathématique, December 1999, Volume 78, Issue 1, pp 61-116.

This criterion has almost nothing to do with integer entries. However, the idea is pretty simple; use the Birkhoff metric on strictly positive matrices.

If we weaken the condition to $M$ merely being primitive but not strictly positive, then there exists a sequence of $A_i$s with no zero rows or zero columns such that $(MA_i)$ is not weakly ergodic. This can probably be arranged with the $A_i$ in $GL(n,Z)$ as well, but I haven't thought about it (much).

In your notation, you are using forward products, while the reference refers to backward products, but the results are equivalent.

Answer 2

I would look at Chapter III of "Products of random matrices" by Bougerol and Lacroix. Probably what you´re looking for is Theorem 4.3 part ii (page 56).

In you're case it would go something like this (it might be necessary to transpose the matrices in the sequence to make them fit the theorem, I can never tell beforehand). Let $B_i = MA_i$.

1. Verify that there is no finite collection of subspaces which are permuted by all possible values of $B_i$ (Strong irreducibility).
2. Verify that there is a positive probability that $B_i$ has a real largest eigenvalue with multiplicity $1$ (Contraction).

It follows from the theorem that there exists a random one dimensional subspace $Z$ such that for all non-zero vectors $x$ one has that $B_1\cdots B_n x$ converges to $Z$ almost surely. Also, if $\mu$ is a probability on the projective space (i.e. on one dimensional subspaces) then $B_1 \cdots B_n \mu$ converges almost surely to the Dirac delta on $Z$.