A representation for a weakly ergodic sequence of (nonstochastic) matrices allows products of nonnegative 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 colum entries assumed by previous studies. Extensions beyond stationarity are discussed.
[1]
H. Furstenberg,et al.
Products of Random Matrices
,
1960
.
[2]
J. Kingman.
Subadditive Ergodic Theory
,
1973
.
[3]
Hiroshi Ishitani,et al.
A Central Limit Theorem for the Subadditive Process and Its Application to Products of Random Matrices
,
1976
.
[4]
J. Hajnal,et al.
On products of non-negative matrices
,
1976,
Mathematical Proceedings of the Cambridge Philosophical Society.
[5]
P. Hall,et al.
Martingale Limit Theory and Its Application
,
1980
.
[6]
R. C. Bradley.
Basic Properties of Strong Mixing Conditions
,
1985
.
[7]
C. Heyde,et al.
Confidence intervals for demographic projections based on products of random matrices.
,
1985,
Theoretical population biology.
[8]
C. Heyde.
An asymptotic representation for products of random matrices
,
1985
.
[9]
Olle Nerman,et al.
On Products of Nonnegative Matrices
,
1990
.