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For a long time the standard (though not the easiest to find) reference on random matrix products was Bougerol and Lacrois:

Bougerol, Philippe, and Jean Lacroix. Products of random matrices with applications to Schrödinger operators. Birkhäuser, 1985

Given, however, that this book is almost thirty years old, I wonder if anyone can recommend a more up-to-date reference?!

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I don't have a textbook reference from this century, but some more recent references than Bougerol & Lacroix are:

A. Crisanti, G. Paladin, and A. Vulpiani, Products of Random Matrices in Statistical Physics, Springer, 1993.

G. Högnäs and A. Mukherjee, Probability Measures on Semigroups: Convolution Products, Random Walks, and Random Matrices, Plenum, 1995.

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