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**Products of random matrices: Convergence theorems.**
*(English)*
Zbl 0592.60015

Random matrices and their applications, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 31-54 (1986).

[For the entire collection see Zbl 0581.00014.]

In this 23 pages long paper the authors manage to give a comprehensive presentation of the classical limit theorems for products of i.i.d. non- singular random \(n\times n\) matrices, i.e. the law of large numbers, the central limit theorem with remainder term estimate, the law of the iterated logarithm, the functional central limit theorem, the large deviation theorem and the renewal theorem.

The paper concentrates on the crucial ideas that have led to these results and for readers with a solid background in mathematics and probability theory interested in understanding the basic ideas of the theory of products of random matrices this paper is excellent. For readers with a little less mathematical background perhaps the recent book ”Products of random matrices with applications to SchrĂ¶dinger operators.” (1985; Zbl 0572.60001) by P. Bougerol and J. Lacroix is better suited.

In this 23 pages long paper the authors manage to give a comprehensive presentation of the classical limit theorems for products of i.i.d. non- singular random \(n\times n\) matrices, i.e. the law of large numbers, the central limit theorem with remainder term estimate, the law of the iterated logarithm, the functional central limit theorem, the large deviation theorem and the renewal theorem.

The paper concentrates on the crucial ideas that have led to these results and for readers with a solid background in mathematics and probability theory interested in understanding the basic ideas of the theory of products of random matrices this paper is excellent. For readers with a little less mathematical background perhaps the recent book ”Products of random matrices with applications to SchrĂ¶dinger operators.” (1985; Zbl 0572.60001) by P. Bougerol and J. Lacroix is better suited.

Reviewer: T.Kaijser

### MSC:

60F05 | Central limit and other weak theorems |

60J05 | Discrete-time Markov processes on general state spaces |

15B52 | Random matrices (algebraic aspects) |