Strong laws of large numbers for products of random matrices

Author:
Steve Pincus

Journal:
Trans. Amer. Math. Soc. **287** (1985), 65-89

MSC:
Primary 60F15; Secondary 60B15

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766207-5

MathSciNet review:
766207

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This work, on products of random matrices, is inspired by papers of Furstenberg and Kesten (Ann. Math. Statist. **31** (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. **108** (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures.

We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are and Bernoulli, i.e., random matrices whose support is two points. Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.

**[K]**Kai Lai Chung,*A course in probability theory*, 2nd ed., Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 21. MR**0346858****[W]**William Feller,*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154****[H]**Harry Furstenberg,*Noncommuting random products*, Trans. Amer. Math. Soc.**108**(1963), 377–428. MR**0163345**, https://doi.org/10.1090/S0002-9947-1963-0163345-0**[H]**H. Furstenberg and H. Kesten,*Products of random matrices*, Ann. Math. Statist.**31**(1960), 457–469. MR**0121828**, https://doi.org/10.1214/aoms/1177705909**[T]**Thomas Kaijser,*A limit theorem for Markov chains in compact metric spaces with applications to products of random matrices*, Duke Math. J.**45**(1978), no. 2, 311–349. MR**0501398****[L]**Lynn H. Loomis,*An introduction to abstract harmonic analysis*, D. Van Nostrand Company, Inc., Toronto-New York-London, 1953. MR**0054173****[V]**V. V. Sazonov and V. N. Tutubalin,*Probability distributions on topological groups*, Teor. Verojatnost. i Primenen.**11**(1966), 3–55 (Russian, with English summary). MR**0199872****[G]**Georgi E. Shilov,*Linear algebra*, Revised English edition. Translated from the Russian and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0276252****[V]**V. N. Tutubalin,*Limit theorems for a product of random matrices*, Teor. Verojatnost. i Primenen.**10**(1965), 19–32 (Russian, with English summary). MR**0175169****1.**V. N. Tutubalin,*On measures whose support is generated by a Lie algebra*, Teor. Verojatnost. i Primenen**12**(1967), 154–160 (Russian, with English summary). MR**0219102****2.**-, (1969),*Some theorems of the type of the strong law of large numbers*, Theory Probab. Appl.**14**, 313-319.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60F15,
60B15

Retrieve articles in all journals with MSC: 60F15, 60B15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766207-5

Keywords:
Random matrices,
Bernoulli matrices

Article copyright:
© Copyright 1985
American Mathematical Society