A representation for a weakly ergodic sequence of (non stochastic) matrices
allows
products of non-negative matrices which
eventually become strictly positive to be expressed via products of some associated
stochastic matrices and ratios of values of a certain
function. This formula used in a random setup leads to a
representation for the logarithm of a random matrix product.
If the sequence of random matrices is in addition stationary then
automatically
almost all sequences are
weakly ergodic, and the
representation is expressed in terms of
an one--dimensional stationary process. This permits properties of products of
random matrices to be deduced
from the latter. Second moment assumptions guarantee that central limit theorems
and laws of the iterated logarithm hold for the random matrix products if
and only if they hold
for the corresponding stationary process. Finally, a central limit theorem for
some classes of weakly dependent
stationary random matrices is derived doing away with the restriction of
boundedness of the ratios of column entries assumed by previous studies.
Extensions beyond stationarity are discussed.